Theorem rexbid 1654

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 1650	1 |- (ph -> (E.x e. A ps <-> E.x e. A ch))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   e. wcel 955  E.wrex 1638

This theorem is referenced by:  rexbidv 1656  rexbii 1660  uniiunlem 2122  iunfi 4543  tz9.13g 4636  scott0 4689  infcvgaux1 7154  homcard 10426  fgsb 10444  fgsb2 10449

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-ex 978  df-rex 1642

Copyright terms: Public domain