Theorem ralbid 1653

Description: Formula-building rule for restricted universal quantifier (deduction rule).

Hypotheses

Ref	Expression
ralbid.1	|- (ph -> A.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 |- (ph -> A.xph)
2		ralbid.2	. . 3 |- (ph -> (ps <-> ch))
3	2	adantr 389	. 2 |- ((ph /\ x e. A) -> (ps <-> ch))
4	1, 3	ralbida 1649	1 |- (ph -> (A.x e. A ps <-> A.x e. A ch))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   e. wcel 955  A.wral 1637

This theorem is referenced by:  ralbidv 1655  ralbii 1659  sbcralt 1980  sbcrext 1981  sbcralgf 1982  sbcrexgf 1983  zfrep6 3600  cplem2 4693  ac6lem 4726  lble 5994  irredt 10230

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1641

Copyright terms: Public domain