Theorem ralbid 2495

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 2491	1 |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1474   e. wcel 2167   A.wral 2475

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 1461  ax-gen 1463  ax-4 1524

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-ral 2480

This theorem is referenced by:  ralbidv  2497  sbcralt  3066  riota5f  5905  mkvprop  7233  lble  8991  ellimc3apf  14980  strcollnft  15714