Theorem ralbid 2464

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 274	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
4	1, 3	ralbida 2460	1 |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1448   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-5 1435  ax-gen 1437  ax-4 1498

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2449

This theorem is referenced by:  ralbidv  2466  sbcralt  3027  riota5f  5822  mkvprop  7122  lble  8842  ellimc3apf  13269  strcollnft  13866