Theorem raleqbii 2478

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 2233	. . 3 |- ( x e. A <-> x e. B )
3		raleqbii.2	. . 3 |- ( ps <-> ch )
4	2, 3	imbi12i 238	. 2 |- ( ( x e. A -> ps ) <-> ( x e. B -> ch ) )
5	4	ralbii2 2476	1 |- ( A. x e. A ps <-> A. x e. B ch )

Syntax hints:   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-clel 2161  df-ral 2449

This theorem is referenced by: (None)