Theorem rexeqbid 3151

Description: Equality deduction for restricted existential quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Hypotheses

Ref	Expression
raleqbid.0	|- F/ x ph
raleqbid.1	|- F/_ x A
raleqbid.2	|- F/_ x B
raleqbid.3	|- ( ph -> A = B )
raleqbid.4	|- ( ph -> ( ps <-> ch ) )

Assertion

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

Proof of Theorem rexeqbid

Step	Hyp	Ref	Expression
1		raleqbid.3	. . 3 |- ( ph -> A = B )
2		raleqbid.1	. . . 4 |- F/_ x A
3		raleqbid.2	. . . 4 |- F/_ x B
4	2, 3	rexeqf 3135	. . 3 |- ( A = B -> ( E. x e. A ps <-> E. x e. B ps ) )
5	1, 4	syl 17	. 2 |- ( ph -> ( E. x e. A ps <-> E. x e. B ps ) )
6		raleqbid.0	. . 3 |- F/ x ph
7		raleqbid.4	. . 3 |- ( ph -> ( ps <-> ch ) )
8	6, 7	rexbid 3051	. 2 |- ( ph -> ( E. x e. B ps <-> E. x e. B ch ) )
9	5, 8	bitrd 268	1 |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   F/wnf 1708   F/_wnfc 2751   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:  iuneq12df  4544