DENNIS JACOBS, Circuit Judge:

Following a jury trial in the United States District Court for the Southern District of New York (Rakoff, J.), Leonard C. Johnson was convicted of (1) bank robbery, (2) armed bank robbery, and (3) using and carrying a firearm during and in connection with a crime of violence — the bank robbery. In a prior appeal, we vacated the bank robbery conviction on the ground of multiplicity with the conviction for armed bank robbery in the same incident. The current 28 U.S.C. § 2255 petition argues that the firearms conviction must be now vacated because there is no subsisting conviction on the predicate crime of violence. The district court (Engelmayer, J.)1 denied the petition, construing our precedents — in particular, United States v. Zhou, 428 F.3d 361 (2d Cir.2005) — to suggest that a defendant can be convicted under 18 U.S.C. § 924(c) without being convicted of the predicate crime in connection with which a firearm was used.

We hold that a conviction under 18 U.S.C. § 924(c) requires legally sufficient proof that the predicate crime of violence (or drug trafficking) was committed, but does not require a conviction for that predicate crime.

Affirmed.

I

In August 1998, Johnson entered a bank, pointed a gun at the cashier, and demanded money. In connection with that single incident, he was charged with (1) bank robbery, in violation of 18 U.S.C. § 2113(a) ("Count One"); (2) armed bank robbery, in violation of 18 U.S.C. § 2113(d) ("Count Two"); and (3) using and carrying a firearm during and in connection with a crime of violence in violation of 18 U.S.C. § 924(c) ("Count Three"). The indictment charged that Johnson committed the firearms offense in Count Three during and in connection with the "bank robbery charged in Count One."

A jury found Johnson guilty on all three counts. Johnson was sentenced to 240 months of imprisonment on Count One and 274 months on Count Two, to run concurrently, and to 300 months on Count Three, to run consecutively. On the direct appeal, Johnson challenged his convictions and sentence on a number of grounds, all of which we rejected. United States v. Johnson, 24 Fed.Appx. 70 (2d Cir.2001).

Johnson then filed a pro se petition for relief under 28 U.S.C. § 2255, arguing (inter alia) that his convictions on Counts One and Two violated the Double Jeopardy Clause.2 Our September 2008 summary order vacated the conviction on Count One on the ground that it should have been merged into the Count Two conviction as a lesser included offense. Johnson v. United States, 293 Fed.Appx. 789, 790 (2d Cir.2008) (citing United States v. Sappe, 898 F.2d 878 (2d Cir.1990)). Johnson's aggregate sentence was unchanged, however, because the sentence on Count Two was greater than the Count One sentence and those sentences were to run concurrently. Id. We rejected the remainder of Johnson's arguments as meritless.

Following a number of procedural developments, Johnson filed the present 28 U.S.C. § 2255 petition — his third — with the assistance of new counsel. This new petition argues (inter alia) that: (1) the conviction on Count Three must be vacated in light of the vacatur of the conviction on Count One, and (2) Johnson's prior counsel provided ineffective assistance. The district court rejected these arguments, but granted a certificate of appealability as to both issues.

The government encourages us to avoid the merits, arguing that Johnson is procedurally barred from asserting his claims, either because he previously failed to present them or because, having previously presented them, he is bound by the law of the case. For his part, Johnson argues that the government waived its procedural arguments by not properly presenting them below, and that any default was excused by the ineffectiveness of prior defense counsel.

We see no reason to undertake a (likely unedifying) tour of the procedural landscape: there are two prior summary orders of this Court, along with an opinion interpreting an intervening Supreme Court case, see Johnson v. United States, 623 F.3d 41, 42 (2d Cir.2010) (interpreting Magwood v. Patterson, 561 U.S. 320, 323-24, 130 S.Ct. 2788, 177 L.Ed.2d 592 (2010)); rulings by two district judges (Judge Rakoff and Judge Engelmayer) and two magistrate judges (Judge Maas and Judge Fox) on three separate 28 U.S.C. § 2255 petitions; and multiple sets of defense counsel, as well as intervals during which Johnson was pro se. "It is well-settled that the doctrine of procedural default is prudential rather than jurisdictional in nature." Kuhali v. Reno, 266 F.3d 93, 101 (2d Cir.2001). And "the law of the case doctrine does not bind us with the same rigidity as it binds the district court." United States v. Tenzer, 213 F.3d 34, 40 (2d Cir.2000).

Johnson's arguments fail on the merits. "We review de novo a district court's denial of a 28 U.S.C. § 2255 petition." Kaminski v. United States, 339 F.3d 84, 86 (2d Cir.2003).

II

The indictment specified that the "bank robbery charged in Count One" was the predicate crime of violence in connection with which Johnson committed the Count Three firearms offense. Johnson argues that his Count Three conviction (which entails a 300-month consecutive sentence) cannot stand now that his Count One conviction has been vacated.3 This argument does not work unless (A) the vacatur (and merger) of the Count One conviction called into question the factual basis for that conviction and, hence, the factual basis for the Count Three conviction, or (B) a conviction on Count One was a legal prerequisite of conviction on the 18 U.S.C. § 924(c) offense charged in Count Three.

A

The first premise — that the vacatur of Count One undermined the factual basis for the Count Three conviction — is unsound for reasons we explained in United States v. Ansaldi:

372 F.3d 118, 125 (2d Cir.2004).

Johnson urges us to overrule Ansaldi. However, a panel of this Court is "bound by the decisions of prior panels until such time as they are overruled either by an en banc panel of our Court or by the Supreme Court." United States v. Wilkerson, 361 F.3d 717, 732 (2d Cir.2004). In any event, Johnson offers no persuasive argument to challenge Ansaldi's holding or its reasoning. We therefore reject the premise that the vacatur of Count One disturbed the factual basis for the conviction on Count Three.

B

Can a defendant be convicted of a firearms offense under 18 U.S.C. § 924(c) who has not been convicted of the predicate crime of violence (or drug trafficking) in connection with which the firearms offense was committed? We have previously discussed that question, but declined to answer it. See Zhou, 428 F.3d at 378-79 & n. 16.

Under 18 U.S.C. § 924(c), "any person who, during and in relation to any crime of violence or drug trafficking crime ... for which the person may be prosecuted in a court of the United States, uses or carries a firearm, or who, in furtherance of any such crime, possesses a firearm, shall" be subject to criminal penalties. 18 U.S.C. § 924(c)(1)(A). Among the crimes of violence that may serve as a predicate for a § 924(c) conviction are the bank robbery and armed bank robbery offenses charged against Johnson in Counts One and Two. 18 U.S.C. § 924(c)(3) (defining "crime of violence" to include any felony that "has as an element the use, attempted use, or threatened use of physical force against the person or property of another"); see 18 U.S.C. § 2113(a), (d).

The Zhou defendants were charged with Hobbs Act extortion and conspiracy to commit extortion, as well as § 924(c) offenses in connection with those extortion-related crimes. 428 F.3d at 364. The defendants challenged the sufficiency of evidence for the extortion-related crimes and argued that their convictions under § 924(c) likewise had to be reversed. We concluded that the proof of extortion was indeed legally insufficient, and that "if the proof of the predicate crime is so fundamentally deficient that it could not sustain a conviction, then likewise, a conviction under § 924 cannot stand." Id. at 377-79. A § 924(c) conviction must be supported by, at the very least, "legally sufficient proof of the underlying offense." Id. at 379.

Zhou expressly declined to resolve two questions: (1) whether a § 924(c) conviction can be sustained if, notwithstanding legally sufficient proof, the jury acquits on the predicate crime, id. at 379 n. 16, and (2) whether a § 924(c) conviction requires that the defendant be charged with and convicted of the predicate crime, id. at 378-79; see also United States v. Parker, 577 F.3d 143, 146 n. 2 (2d Cir.2009).4 We need not answer the first question — on which there is arguably a circuit split — because the jury found beyond a reasonable doubt that Johnson committed both bank robbery (Count One) and armed bank robbery (Count Two). Compare United States v. Ruiz, 986 F.2d 905, 911 (5th Cir.1993) ("Because there is more than ample evidence showing that a reasonable jury could have found [the defendant] guilty of [the predicate crime], the fact that [the defendant] was acquitted on that count does not preclude his conviction under Section 924(c)(1)."), with United States v. Carter, 300 F.3d 415, 425 (4th Cir.2002) (disapproving the "extreme position" in Ruiz).

The second question, however, must be decided now because Johnson was convicted under § 924(c) of using a firearm in connection with the Count One robbery, and the Count One conviction was vacated.5 Every circuit court to have considered the issue has concluded that § 924(c) does not require the defendant to be convicted of (or even charged with) the predicate crime, so long as there is legally sufficient proof that the predicate crime was, in fact, committed. United States v. Lake, 150 F.3d 269, 275 (3d Cir.1998); Carter, 300 F.3d at 425; United States v. Munoz-Fabela, 896 F.2d 908, 911 (5th Cir. 1990); United States v. Nelson, 27 F.3d 199, 200 (6th Cir.1994); Myers v. United States, 993 F.2d 171, 172 (8th Cir.1993) (per curiam); United States v. Hunter, 887 F.2d 1001, 1003 (9th Cir.1989); United States v. Hill, 971 F.2d 1461, 1467 (10th Cir.1992) (en banc); United States v. Frye, 402 F.3d 1123, 1127-28 (11th Cir.2005) (per curiam).

We join that consensus. The plain language of § 924(c) requires only that the predicate crime of violence (or drug trafficking) have been committed; the wording does not suggest that the defendant must be separately charged with that predicate crime and be convicted of it. See 18 U.S.C. § 924(c)(1)(A) ("[A]ny person who, during and in relation to any crime of violence or drug trafficking crime ... for which the person may be prosecuted in a court of the United States, uses or carries a firearm, or who, in furtherance of any such crime, possesses a firearm, shall [be subject to criminal penalties]."). In this case, there was legally sufficient proof that Johnson committed bank robbery: the jury returned a guilty verdict on Count One. The vacatur of the Count One conviction on grounds unrelated to sufficiency did not undermine the conviction on Count Three.

III

"A claim of ineffective assistance entails a showing that: 1) the defense counsel's performance was objectively unreasonable; and 2) the deficient performance prejudiced the defense." Kovacs v. United States, 744 F.3d 44, 49 (2d Cir. 2014) (citing Strickland v. Washington, 466 U.S. 668, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984)). Johnson claims that prior counsel was ineffective for failing to (1) identify the multiplicity argument that resulted in the vacatur of Count One, and (2) challenge Count Three on the grounds that Johnson advances now.

Neither claim succeeds: (1) To the extent counsel was deficient in failing to challenge Count One, any possible prejudice has already been remedied by our vacatur of Count One,6 see Johnson, 293 Fed.Appx. at 790; and (2) counsel's failure to challenge Count Three based on the arguments Johnson advances now was not objectively unreasonable, since those arguments are, for reasons discussed above, meritless, see United States v. Kirsh, 54 F.3d 1062, 1071 (2d Cir.1995).

CONCLUSION

For the foregoing reasons, we affirm the order of the district court.