Theorem eqrdv

Description: Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996)

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

Proof

Step	Hyp	Ref	Expression
1		eqrdv.1	|- ( ph -> ( x e. A <-> x e. B ) )
2	1	alrimiv	|- ( ph -> A. x ( x e. A <-> x e. B ) )
3		dfcleq	|- ( A = B <-> A. x ( x e. A <-> x e. B ) )
4	2 3	sylibr	|- ( ph -> A = B )