Theorem ralbid 2531

Description: Formula-building rule for restricted universal quantifier (deduction form). (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 276	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
4	1, 3	ralbida 2527	1 |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1509   e. wcel 2202   A.wral 2511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-4 1559

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2516

This theorem is referenced by:  ralbidv  2533  sbcralt  3109  riota5f  6008  mkvprop  7417  lble  9186  ellimc3apf  15471  strcollnft  16700