Theorem ralbid 2633

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

Hypotheses

Ref	Expression
ralbid.1	|- F/xph
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/xph
2		ralbid.2	. . 3 |- (ph -> (ps <-> ch))
3	2	adantr 451	. 2 |- ((ph /\ x e. A) -> (ps <-> ch))
4	1, 3	ralbida 2629	1 |- (ph -> (A.x e. A ps <-> A.x e. A ch))

Syntax hints:   -> wi 4   <-> wb 176   F/wnf 1544   e. wcel 1710  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  ralbidv  2635  sbcralt  3119  sbcrext  3120