Theorem bireudva 1317

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

Hypothesis

Ref	Expression
bireudva.1	|- ((ph /\ x e. A) -> (ps <-> ch))

Assertion

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

Distinct variable group(s):   ph,x

Proof of Theorem bireudva

Step	Hyp	Ref	Expression
1		bireudva.1	. . . . 5 |- ((ph /\ x e. A) -> (ps <-> ch))
2	1	exp 291	. . . 4 |- (ph -> (x e. A -> (ps <-> ch)))
3	2	pm5.32d 491	. . 3 |- (ph -> ((x e. A /\ ps) <-> (x e. A /\ ch)))
4	3	bieudv 1013	. 2 |- (ph -> (E!x(x e. A /\ ps) <-> E!x(x e. A /\ ch)))
5		df-reu 1207	. 2 |- (E!x e. A ps <-> E!x(x e. A /\ ps))
6		df-reu 1207	. 2 |- (E!x e. A ch <-> E!x(x e. A /\ ch))
7	4, 5, 6	3bitr4g 428	1 |- (ph -> (E!x e. A ps <-> E!x e. A ch))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E!weu 1007   e. wcel 1092  E!wreu 1203

This theorem is referenced by:  bireudv 1318  zmax 4618  zbtwnre 4619  rebtwnz 4620

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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