Theorem rexeqbii 2563

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
raleqbii.1	|- A = B
raleqbii.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		raleqbii.1	. . . 4 |- A = B
2	1	eleq2i 2305	. . 3 |- ( x e. A <-> x e. B )
3		raleqbii.2	. . 3 |- ( ps <-> ch )
4	2, 3	anbi12i 464	. 2 |- ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) )
5	4	rexbii2 2561	1 |- ( E. x e. A ps <-> E. x e. B ch )

Syntax hints:   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-rex 2534

This theorem is referenced by:  exmidsbthrlem  16972