Theorem raleqbii 2990

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 2693	. . 3 |- ( x e. A <-> x e. B )
3		raleqbii.2	. . 3 |- ( ps <-> ch )
4	2, 3	imbi12i 340	. 2 |- ( ( x e. A -> ps ) <-> ( x e. B -> ch ) )
5	4	ralbii2 2978	1 |- ( A. x e. A ps <-> A. x e. B ch )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912

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-ral 2917

This theorem is referenced by:  wfrlem5  7419  ply1coe  19666  ordtbaslem  20992  iscusp2  22106  isrgr  26455  frrlem5  31784  elghomOLD  33686  iscrngo2  33796  tendoset  36047  comptiunov2i  37998