Theorem raleqbidva 2675

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 2462	. 2 |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )
3		raleqbidva.1	. . 3 |- ( ph -> A = B )
4	3	raleqdv 2667	. 2 |- ( ph -> ( A. x e. A ch <-> A. x e. B ch ) )
5	2, 4	bitrd 187	1 |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> 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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449

This theorem is referenced by: (None)