Theorem raleqbidva 3154

Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)

Hypotheses

Ref	Expression
raleqbidva.1	|- ( ph -> A = B )
raleqbidva.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
raleqbidva	|- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hints:   ps( x)   ch( x)

Proof of Theorem raleqbidva

Step	Hyp	Ref	Expression
1		raleqbidva.2	. . 3 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
2	1	ralbidva 2985	. 2 |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )
3		raleqbidva.1	. . 3 |- ( ph -> A = B )
4	3	raleqdv 3144	. 2 |- ( ph -> ( A. x e. A ch <-> A. x e. B ch ) )
5	2, 4	bitrd 268	1 |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = 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-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917

This theorem is referenced by:  catpropd  16369  cidpropd  16370  funcpropd  16560  fullpropd  16580  natpropd  16636  gsumpropd2lem  17273  istrkgc  25353  istrkgb  25354  istrkgcb  25355  istrkge  25356  iscgrg  25407  isperp  25607  clwlkclwwlk  26903  rngurd  29788  matunitlindflem1  33405