Theorem rexeqbii 2510

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 2263	. . 3 |- ( x e. A <-> x e. B )
3		raleqbii.2	. . 3 |- ( ps <-> ch )
4	2, 3	anbi12i 460	. 2 |- ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) )
5	4	rexbii2 2508	1 |- ( E. x e. A ps <-> E. x e. B ch )

Syntax hints:   <-> wb 105   = wceq 1364   e. wcel 2167   E.wrex 2476

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-rex 2481

This theorem is referenced by:  exmidsbthrlem  15666