Theorem eqrdav 2164

Description: Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.)

Hypotheses

Ref	Expression
eqrdav.1	|- ( ( ph /\ x e. A ) -> x e. C )
eqrdav.2	|- ( ( ph /\ x e. B ) -> x e. C )
eqrdav.3	|- ( ( ph /\ x e. C ) -> ( x e. A <-> x e. B ) )

Assertion

Ref	Expression
eqrdav	|- ( ph -> A = B )

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hint:   C( x)

Proof of Theorem eqrdav

Step	Hyp	Ref	Expression
1		eqrdav.1	. . . 4 |- ( ( ph /\ x e. A ) -> x e. C )
2		eqrdav.3	. . . . . 6 |- ( ( ph /\ x e. C ) -> ( x e. A <-> x e. B ) )
3	2	biimpd 143	. . . . 5 |- ( ( ph /\ x e. C ) -> ( x e. A -> x e. B ) )
4	3	impancom 258	. . . 4 |- ( ( ph /\ x e. A ) -> ( x e. C -> x e. B ) )
5	1, 4	mpd 13	. . 3 |- ( ( ph /\ x e. A ) -> x e. B )
6		eqrdav.2	. . . 4 |- ( ( ph /\ x e. B ) -> x e. C )
7	2	exbiri 380	. . . . . 6 |- ( ph -> ( x e. C -> ( x e. B -> x e. A ) ) )
8	7	com23 78	. . . . 5 |- ( ph -> ( x e. B -> ( x e. C -> x e. A ) ) )
9	8	imp 123	. . . 4 |- ( ( ph /\ x e. B ) -> ( x e. C -> x e. A ) )
10	6, 9	mpd 13	. . 3 |- ( ( ph /\ x e. B ) -> x e. A )
11	5, 10	impbida 586	. 2 |- ( ph -> ( x e. A <-> x e. B ) )
12	11	eqrdv 2163	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-17 1514  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158

This theorem is referenced by:  supminfex  9535  fzdifsuc  10016