Theorem rexbid 1662

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

Hypotheses

Ref	Expression
ralbid.1	|- (ph -> A.xph)
ralbid.2	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
rexbid	|- (ph -> (E.x e. A ps <-> E.x e. A ch))

Proof of Theorem rexbid

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	rexbida 1658	1 |- (ph -> (E.x e. A ps <-> E.x e. A ch))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   e. wcel 958  E.wrex 1646

This theorem is referenced by:  rexbidv 1664  rexbii 1668  uniiunlem 2132  iunfiOLD 4569  tz9.13g 4664  scott0 4717  infcvgaux1 7219  homcard 10539  fgsb 10570  fgsbOLD 10571  fgsb2 10580

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-rex 1650

Copyright terms: Public domain