Theorem reubidva 1776

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

Hypothesis

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

Assertion

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

Distinct variable group:   ph,x

Proof of Theorem reubidva

Step	Hyp	Ref	Expression
1		reubidva.1	. . . 4 |- ((ph /\ x e. A) -> (ps <-> ch))
2	1	pm5.32da 648	. . 3 |- (ph -> ((x e. A /\ ps) <-> (x e. A /\ ch)))
3	2	eubidv 1384	. 2 |- (ph -> (E!x(x e. A /\ ps) <-> E!x(x e. A /\ ch)))
4		df-reu 1648	. 2 |- (E!x e. A ps <-> E!x(x e. A /\ ps))
5		df-reu 1648	. 2 |- (E!x e. A ch <-> E!x(x e. A /\ ch))
6	3, 4, 5	3bitr4g 554	1 |- (ph -> (E!x e. A ps <-> E!x e. A ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 956  E!weu 1378  E!wreu 1644

This theorem is referenced by:  reubidv 1777  exfo 3813  f1ofveu 3873  zmax 6176  zbtwnre 6177  rebtwnz 6178

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-17 969  ax-4 971  ax-5o 973

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-eu 1380  df-reu 1648

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