Theorem rexeqbidva 3155

Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)

Hypotheses

Ref	Expression
raleqbidva.1	|- ( ph -> A = B )
raleqbidva.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
rexeqbidva	|- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )

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

Allowed substitution hints:   ps( x)   ch( x)

Proof of Theorem rexeqbidva

Step	Hyp	Ref	Expression
1		raleqbidva.2	. . 3 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
2	1	rexbidva 3049	. 2 |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )
3		raleqbidva.1	. . 3 |- ( ph -> A = B )
4	3	rexeqdv 3145	. 2 |- ( ph -> ( E. x e. A ch <-> E. x e. B ch ) )
5	2, 4	bitrd 268	1 |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918

This theorem is referenced by:  catpropd  16369  istrkgb  25354  istrkgcb  25355  istrkge  25356  isperp  25607  perpcom  25608  eengtrkg  25865  eengtrkge  25866  afsval  30749  matunitlindflem2  33406