Theorem ralbid 2366

Description: Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)

Hypotheses

Ref	Expression
ralbid.1	|- F/ x ph
ralbid.2	|- ( ph -> ( ps <-> ch ) )

Assertion

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

Proof of Theorem ralbid

Step	Hyp	Ref	Expression
1		ralbid.1	. 2 |- F/ x ph
2		ralbid.2	. . 3 |- ( ph -> ( ps <-> ch ) )
3	2	adantr 270	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
4	1, 3	ralbida 2362	1 |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 103   F/wnf 1389   e. wcel 1433   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-4 1440

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-ral 2353

This theorem is referenced by:  ralbidv  2368  sbcralt  2890  riota5f  5512  lble  8025