Metamath Proof Explorer

Theorem eqriv

Description: Infer equality of classes from equivalence of membership. (Contributed by NM, 21-Jun-1993)

Ref Expression
Hypothesis eqriv.1 
|- ( x e. A <-> x e. B )
Assertion eqriv 
|- A = B

Proof

Step Hyp Ref Expression
1 eqriv.1 
 |-  ( x e. A <-> x e. B )
2 dfcleq 
 |-  ( A = B <-> A. x ( x e. A <-> x e. B ) )
3 2 1 mpgbir 
 |-  A = B