Theorem reubidva 1779

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 649	. . 3 |- (ph -> ((x e. A /\ ps) <-> (x e. A /\ ch)))
3	2	eubidv 1386	. 2 |- (ph -> (E!x(x e. A /\ ps) <-> E!x(x e. A /\ ch)))
4		df-reu 1651	. 2 |- (E!x e. A ps <-> E!x(x e. A /\ ps))
5		df-reu 1651	. 2 |- (E!x e. A ch <-> E!x(x e. A /\ ch))
6	3, 4, 5	3bitr4g 555	1 |- (ph -> (E!x e. A ps <-> E!x e. A ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 958  E!weu 1380  E!wreu 1647

This theorem is referenced by:  reubidv 1780  exfo 3822  f1ofveu 3882  zmax 6220  zbtwnre 6221  rebtwnz 6222

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-eu 1382  df-reu 1651

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