Theorem rexeqbii 3054

Description: Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
rexeqbii.1	|- A = B
rexeqbii.2	|- ( ps <-> ch )

Assertion

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

Proof of Theorem rexeqbii

Step	Hyp	Ref	Expression
1		rexeqbii.1	. . . 4 |- A = B
2	1	eleq2i 2693	. . 3 |- ( x e. A <-> x e. B )
3		rexeqbii.2	. . 3 |- ( ps <-> ch )
4	2, 3	anbi12i 733	. 2 |- ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) )
5	4	rexbii2 3039	1 |- ( E. x e. A ps <-> E. x e. B ch )

Syntax hints:   <-> wb 196   = 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-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-rex 2918

This theorem is referenced by:  bnj882  30996