Theorem bireudv 1318

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

Hypothesis

Ref	Expression
bireudv.1	|- (ph -> (ps <-> ch))

Assertion

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

Distinct variable group(s):   ph,x

Proof of Theorem bireudv

Step	Hyp	Ref	Expression
1		bireudv.1	. . 3 |- (ph -> (ps <-> ch))
2	1	adantr 306	. 2 |- ((ph /\ x e. A) -> (ps <-> ch))
3	2	bireudva 1317	1 |- (ph -> (E!x e. A ps <-> E!x e. A ch))

Syntax hints:   -> wi 2   <-> wb 127   e. wcel 1092  E!wreu 1203

This theorem is referenced by:  reueqd 1329  oawordeu 3157  aceq6b 3565

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  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-eu 1009  df-reu 1207

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