Theorem birexd 1218

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

Hypotheses

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

Assertion

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

Proof of Theorem birexd

Step	Hyp	Ref	Expression
1		birald.1	. 2 |- (ph -> A.xph)
2		birald.2	. . 3 |- (ph -> (ps <-> ch))
3	2	adantr 306	. 2 |- ((ph /\ x e. A) -> (ps <-> ch))
4	1, 3	birexda 1214	1 |- (ph -> (E.x e. A ps <-> E.x e. A ch))

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  birexdv 1220  birex 1224  tz9.13g 3508  scott0 3542

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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