Boggs, Chief Justice. At a jury trial in 2017, Erasmus Chandler was found guilty of aggravated child molestation and two counts of child molestation. In 2019, the Court of Appeals affirmed his convictions in an unpublished opinion. Chandler later filed a pro se petition for habeas corpus, which the habeas court granted in 2022 on the ground that Chandler’s appellate counsel provided ineffective assistance at the motion for new trial stage and on appeal, including by failing to raise and prove claims of ineffective assistance of trial counsel. Warden Aimee Smith appeals, arguing that the habeas court erred in admitting an exhibit at the habeas hearing and in determining that Chandler’s appellate counsel provided ineffective assistance.[1] At the habeas hearing, the warden did not object to the admission of the challenged exhibit for the limited purpose for which it was admitted, and now on appeal the warden has not shown plain error in its admission. Moreover, the habeas court properly determined that Chandler’s appellate counsel provided ineffective assistance at the motion for new trial stage and on appeal by failing to raise and prove a claim of ineffective assistance of trial counsel for failing to impeach the alleged victim’s testimony at trial with evidence that she had made prior inconsistent statements about the alleged abuse. We affirm the habeas court’s grant of relief on this basis. 1. The record shows as follows. In September 2014, Chandler and his live-in girlfriend, Christina Williams, moved with Williams’ three daughters and Chandler and Williams’ two younger children from Montgomery, Alabama, to Augusta, Georgia, where Williams worked at a cell phone store and Chandler cared for the children. In February 2015, Chandler and Williams drove with the children to Montgomery to visit family and to celebrate the fourteenth birthday of Williams’ oldest daughter, N.C. During the trip, when Williams was telling N.C. and another daughter that they needed to do a better job with their chores, N.C. told Williams that Chandler had been “touching on” N.C. for years and that N.C. thought that she might be pregnant by Chandler. After Williams confronted Chandler, Williams, Chandler, and N.C. got into the family van, where Williams had N.C. repeat the allegations to Chandler. Chandler angrily denied the accusations and demanded that they take N.C. to a hospital immediately. The next morning, Williams drove back to Augusta with the children and took N.C. to Doctors Hospital of Augusta, where the medical staff contacted law enforcement and determined that N.C. was not pregnant. Several days later, Denise Field conducted a forensic interview of N.C., which was played for the jury at Chandler’s trial. During the interview, N.C. told Field that she was in special education classes and that Chandler had licked her vagina and breasts and put something inside of her vagina while he was on top of her. On August 4, 2015, a Richmond County grand jury indicted Chandler for aggravated child molestation by placing his mouth on N.C.’s vagina and two counts of child molestation by placing an unknown object in N.C.’s vagina and by placing his mouth on N.C.’s breast. Chandler was subsequently arrested. At Chandler’s trial in 2017, Williams testified that N.C. had a learning disability, was in special needs classes, read below her grade level, and developed behavioral issues, including extreme anger and suicidal thoughts, around the age of nine or ten, the timeframe when N.C. said that Chandler began abusing her. Williams acknowledged that N.C. sometimes lied about “petty things” and that N.C. wrote in her diary about how much she hated Chandler, did not want Williams to marry him, and wanted a new family but did not write anything about the alleged inappropriate behavior by Chandler aside from an entry that N.C. “scribbled real quick” after the allegations arose but dated to a time before the birthday trip to Montgomery. Williams also acknowledged that she at one point had doubts about N.C.’s allegations due to the hastily scribbled diary entry but later came to believe the allegations because N.C.’s statements had been “very consistent” over time. Williams testified that Chandler admitted that he helped N.C. apply cream to a bump on N.C.’s vagina when N.C. was 13 and Williams was at work. Williams further testified that N.C. had complained in the past about Chandler coming into her bedroom and looking at her when she was seven or eight years old. N.C. testified that Chandler had licked her vagina and breasts and put something inside her vagina when he was on top of her, and Field testified about N.C.’s forensic interview and the disclosure process for children who have been sexually abused. Chandler testified at trial and adamantly denied all the allegations against him, including Williams’ claim that he admitted touching N.C.’s vagina to apply cream to a bump. Chandler also called Officer Jacob Green of the Richmond County Sheriff’s Office, who spoke to Williams and N.C. at the hospital. However, Officer Green was not allowed to testify about what N.C. said, because the trial court sustained the State’s hearsay objection on the ground that Chandler’s counsel did not file a notice of intent to introduce child hearsay. The jury found Chandler guilty of all charges, and he was sentenced to serve a total of 50 years in prison followed by life on probation. Chandler filed a motion for new trial, which he amended with new appellate counsel, claiming that the evidence was insufficient to support his convictions and that the trial court erred in allowing Williams to testify that, although she at one point had doubts about N.C.’s allegations due to the diary entry that N.C. “scribbled real quick” after the allegations arose, she later came to believe them because N.C.’s statements had been “very consistent” over time. After a hearing at which Chandler did not produce any evidence and instead presented only argument, the trial court denied the motion. Chandler, represented by the same counsel, appealed, again raising the two claims that he raised in his amended motion for new trial. On May 21, 2019, the Court of Appeals issued an unpublished opinion rejecting Chandler’s sufficiency claim, finding no plain error from improper bolstering, and affirming the trial court’s judgment. On November 8, 2019, Chandler filed a pro se petition for habeas corpus, which he later amended, raising several claims of ineffective assistance of appellate counsel. The habeas court held an evidentiary hearing over two days in early 2021 at which Chandler’s appellate counsel, James Rogers, and his trial counsel, Sean Gamble, both testified. Chandler introduced into evidence Habeas Exhibit 10, a page from N.C.’s medical records that included a note from Nurse Angela A. Haustad that said: “Pt. told officer that she has only touched [sic] by step father no sexual penetration occurred, told officer that they fight often.” Chandler also introduced Habeas Exhibit 14, another page from N.C.’s medical records, which included a note from Dr. Thomas L. Zickgraf that said that N.C. was “unsure if she has been sexually penetrated by [Chandler's] penis in the past vaginally” and that she “denies any oral contact.”[2] On July 28, 2022, the habeas court entered a lengthy Final Order Granting Habeas Corpus Relief. The habeas court found that both Gamble and Rogers failed to grasp the importance of these notes in N.C.’s medical records. The habeas court determined that Gamble was professionally deficient for, among other things, failing to impeach N.C. by cross-examining her about her prior inconsistent statements contained in Habeas Exhibit 10 and Habeas Exhibit 14 and, if she denied or claimed not to remember making them, failing to introduce the exhibits. The habeas court also determined that this deficient performance prejudiced Chandler, because “[t]he jury was completely unaware of the alleged victim ever being inconsistent or denying the allegations that she made against” Chandler, and the outcome of the case hinged on N.C.’s credibility.[3] The habeas court further determined that Rogers was professionally deficient at the motion for new trial stage and on appeal for, among other things, failing to raise and prove this claim of ineffective assistance of trial counsel in addition to the two weaker issues that Rogers decided to raise. The habeas court also determined that, but for Rogers’ deficient performance, there is a reasonable probability that the outcome of his appeal would have been different. The warden filed a timely notice of appeal. 2. The warden contends that the habeas court erred when it admitted Habeas Exhibit 10 over her hearsay objection, because N.C.’s statements within the document were inadmissible hearsay. However, at the habeas hearing, although the warden objected to the admission of Habeas Exhibit 10 to prove the truth of the statements therein, she said that she had “no objection” to the admission of the exhibit for “the limited purpose” of showing whether Gamble was aware of the document and how he used it in his representation of Chandler. The habeas court then admitted Habeas Exhibit 10 “for that purpose.” The warden now argues that the exhibit should not have been admitted at all. The warden did not make this argument in the habeas court, so we review the habeas court’s ruling only for plain error. Cf. Crayton v. State, 298 Ga. 792, 799 (784 SE2d 343) (2016) (reviewing only for plain error where counsel stated that he had no objection to the admission of documentary evidence). To show plain error, the warden must point to a legal error that was not affirmatively waived, was obvious beyond reasonable dispute, likely affected the outcome of the proceedings, and seriously affected the fairness, integrity, or public reputation of judicial proceedings. See Lupoe v. State, 300 Ga. 233, 243 (794 SE2d 67) (2016). The failure to establish any one of these elements is fatal to the warden’s plain error claim. See Wright v. State, 315 Ga. 459, 462 (883 SE2d 294) (2023). We conclude that the warden has failed to point to a clear legal error by the habeas court. OCGA § 24-8-801 (c) defines “hearsay” as an out-of-court statement “offered in evidence to prove the truth of the matter asserted” in the statement. The habeas court admitted Habeas Exhibit 10 for the limited purpose of showing whether Gamble was aware of it and how he used it in his representation of Chandler, not to prove the truth of the statements that N.C. had “only [been] touched” by Chandler or that “no sexual penetration occurred.” Accordingly, the warden has failed to show error, much less plain error, in the habeas court’s admission of the exhibit over her hearsay objection. 3. The warden also contends that the habeas court erred in determining that Chandler’s appellate counsel was ineffective. “In reviewing the grant or denial of a petition for habeas corpus, this Court accepts the habeas court’s factual findings and credibility determinations unless they are clearly erroneous, but we independently apply the law to the facts.” Luckie v. Berry, 305 Ga. 684, 691 (827 SE2d 644) (2019) (cleaned up). (a) To prevail on a claim of ineffective assistance of appellate counsel, a habeas petitioner must show that his appellate counsel’s performance was deficient and that the deficiency prejudiced the outcome of his appeal. See Cartwright v. Caldwell, 305 Ga. 371, 378 (825 SE2d 168) (2019). See also Strickland v. Washington, 466 U.S. 668, 687 (104 SCt 2052, 80 LE2d 674) (1984). To establish deficient performance, the petitioner must show that his appellate counsel performed his duties in an objectively unreasonable way, considering all the circumstances at the time and in the light of prevailing professional norms. See Cartwright, 305 Ga. at 378. To establish the required prejudice, the petitioner must show that, but for his appellate counsel’s unprofessional errors, there is a reasonable probability that the result of his appeal would have been more favorable. See id. “A reasonable probability is a probability sufficient to undermine confidence in the outcome.” Strickland, 466 U.S. at 694. When the petitioner contends that his appellate counsel performed deficiently by failing to properly raise or prove a claim of ineffective assistance of trial counsel, in order to establish the required prejudice, “the petitioner must demonstrate that the underlying ineffectiveness-of-trial-counsel claim would have had a reasonable probability of success.” Cartwright, 305 Ga. at 378. In other words, to establish the prejudice required to prevail on this type of ineffective assistance of appellate counsel claim, a habeas petitioner must show that his trial counsel’s performance was professionally deficient and that, but for the deficiency, there is a reasonable probability that the outcome of his trial would have been more favorable. See Gramiak v. Beasley, 304 Ga. 512, 513 (820 SE2d 50) (2018). Accordingly, we turn first to whether Chandler was denied the effective assistance of counsel at trial. (b) At Chandler’s trial, N.C. testified that Chandler had licked her vagina and breasts and put something inside her vagina when he was on top of her, and these alleged acts were the basis for the charges of child molestation and aggravated child molestation against Chandler. But according to N.C.’s medical records, she made statements at the hospital that she had “only [been] touched” by Chandler, that “no sexual penetration occurred,” and “den[ying] any oral contact.” Gamble briefly cross-examined N.C. at trial but did not ask her a single question about her statements at the hospital as reflected in the medical records, even though those statements likely would have been admissible as prior inconsistent statements. See Nicholson v. State, 307 Ga. 466, 472 (837 SE2d 362) (2019) (“A prior inconsistent statement of a witness who takes the stand and is subject to cross-examination is admissible as substantive evidence.” (cleaned up)). Moreover, if, on being confronted with the statements by Gamble, N.C. had denied or claimed not to remember making them, then Habeas Exhibit 10 and Habeas Exhibit 14 would have been admissible as extrinsic evidence of the prior inconsistent statements. See OCGA § 24-6-613 (b) (providing for the admission of extrinsic evidence of a prior inconsistent statement if “the witness is first afforded an opportunity to explain or deny the prior inconsistent statement and the opposite party is afforded an opportunity to interrogate the witness on the prior inconsistent statement”). See also OCGA § 24-8-801 (d) (1) (A) (excluding such statements from the definition of hearsay if the declarant testifies at trial and is subject to cross-examination concerning the statement). At the habeas hearing, Gamble testified that his defense strategy was to show the jury that N.C. “was lying, and she wasn’t trustworthy, and she didn’t like [Chandler], and she was making this up because she didn’t like [Chandler].” Impeaching N.C. with her own prior statements and denials as reflected in the medical records would have strongly supported the defense strategy by putting evidence before the jury that she had been significantly inconsistent in her allegations of abuse, information that the jury did not otherwise have. Although the scope of cross-examination will rarely support a claim of deficient performance, under these circumstances, no reasonably competent defense attorney would have decided against presenting this impeachment evidence to cast doubt on the credibility of the State’s key witness. See Cartwright, 305 Ga. at 379. In order to show prejudice, Chandler was not required to show that Gamble’s failure to use the impeachment evidence “more likely than not altered the outcome of the case,” only that “the likelihood of a result more favorable” to him is great enough “to undermine confidence in the outcome.” Strickland, 466 U.S. at 693-695. Chandler made that showing here. The entire case against Chandler was one built on N.C.’s statements about her alleged abuse, and on N.C.’s credibility. No physical evidence supported N.C.’s allegations. All the State’s evidence that the charged crimes had occurred traced back to statements made by N.C. The testimony of N.C., and of Williams and Field about what N.C. told them, was certainly sufficient to support Chandler’s convictions, see Jackson v. Virginia, 443 U.S. 307, 319 (99 SCt 2781, 61 LE2d 560) (1979); OCGA § 24-14-8 (“The testimony of a single witness is generally sufficient to establish a fact.”). But the evidence of Chandler’s guilt was not overwhelming, particularly in light of his adamant denials of N.C.’s accusations; Williams’ testimony that N.C. sometimes lied and hated Chandler; and N.C.’s creation, after she made the accusations, of a backdated entry in her diary that for the first time talked about the things that she claimed that Chandler did to her. See Strickland, 466 U.S. at 696 (“[A] verdict or conclusion only weakly supported by the record is more likely to have been affected by errors than one with overwhelming record support.”). Had the jury been presented with N.C.’s prior inconsistent statements and denials of the alleged abuse, there is a reasonable probability that the outcome of the trial would have been different. The jury heard evidence that N.C. was sometimes untruthful, but it did not hear that, according to her medical records, she had been significantly inconsistent in the very allegations of abuse that formed the basis for the charges against Chandler. To be sure, despite the habeas court’s statement that the jury was “completely unaware of the alleged victim ever being inconsistent” in her allegations against Chandler, there was some evidence of minor inconsistencies about the details of the abuse, but those inconsistencies were not material to the habeas court’s conclusion that Gamble’s deficient performance prejudiced the defense.[4]Presented with N.C.’s prior denials, jurors may well have concluded that N.C.’s trial testimony against Chandler, like the hastily scribbled entry in her diary, was a deliberate fabrication designed to harm Chandler. Moreover, without the impeachment evidence, it was just Chandler’s testimony against that of N.C., Williams, and Field. We are not confident that the jury would have reached the same result if presented with this impeachment evidence. See id. at 694 (“A reasonable probability is a probability sufficient to undermine confidence in the outcome.”). Thus, Chandler has shown Strickland prejudice from Gamble’s deficient performance at trial. The warden resists this conclusion, arguing that in order to show prejudice from Gamble’s deficient performance, Chandler was required to call N.C. to testify at the habeas hearing or to present an appropriate substitute for her sworn testimony such as an affidavit to show how she would have responded when confronted with her prior inconsistent statements. The important point for purposes of assessing prejudice here is that Gamble’s failure to introduce the prior inconsistent statements deprived Chandler of the only evidence of N.C. denying the allegations at issue, which would have been support different in kind from any other evidence that he had to show that she was lying and untrustworthy. Whatever N.C. might have testified on cross-examination could not have had the effect of erasing that evidence entirely: even if she could have cast doubt on or contested the accuracy of the statements, they remained the only evidence of her denying the allegations at issue and thus were still of substantial importance to a defense grounded in trying to discredit N.C. Under these circumstances, calling N.C. to testify at the habeas hearing was not necessary to establish prejudice. See Cartwright, 305 Ga. at 379-381 (reversing denial of habeas relief based on counsel’s failure to impeach a key State witness with evidence of a prior inconsistent statement, despite the lack of testimony or an affidavit from the witness showing how he would have responded to the impeachment evidence, given the less than overwhelming evidence of guilt, counsel’s chosen theory of defense, and the importance of the witness’ testimony to the State’s case). (c) Because Chandler has shown that his trial counsel provided ineffective assistance, any deficiency in his appellate counsel’s failure to raise and prove that ineffectiveness-of-trial- counsel claim prejudiced his appeal. See id. at 381; Gramiak, 304 Ga. at 513. Thus, the only remaining question is whether Rogers was professionally deficient in failing to raise and prove a claim that Gamble provided ineffective assistance by failing to impeach N.C.’s testimony at trial with her prior inconsistent statements contained in Habeas Exhibit 10 and Habeas Exhibit 14. We fail to see why a competent appellate attorney would have failed to raise and support such a claim under these circumstances. As the habeas court recognized, this claim was clearly stronger than the claims of insufficient evidence and improper bolstering that Rogers chose to raise at the motion for new trial stage and on appeal, which were easily rejected. No reasonable attorney would have failed to raise an ineffective-assistance-of-trial-counsel claim based on Gamble’s failure to impeach N.C. with her prior inconsistent statements and denials as reflected in the medical records and to support that claim by presenting Habeas Exhibit 10 and Habeas Exhibit 14 at the motion for new trial hearing, which were essential to proving Gamble’s ineffectiveness. We therefore conclude that Rogers provided ineffective assistance of appellate counsel in this regard, and we affirm the habeas court’s grant of relief on this basis. Judgment affirmed. All the Justices concur, except McMillian, J., disqualified.