Theorem raleqbii 2336

Description: Equality deduction for restricted universal 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
raleqbii	|- ( A. x e. A ps <-> A. x e. B ch )

Proof of Theorem raleqbii

Step	Hyp	Ref	Expression
1		raleqbii.1	. . . 4 |- A = B
2	1	eleq2i 2104	. . 3 |- ( x e. A <-> x e. B )
3		raleqbii.2	. . 3 |- ( ps <-> ch )
4	2, 3	imbi12i 228	. 2 |- ( ( x e. A -> ps ) <-> ( x e. B -> ch ) )
5	4	ralbii2 2334	1 |- ( A. x e. A ps <-> A. x e. B ch )

Syntax hints:   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-ral 2311

This theorem is referenced by: (None)