Theorem reubidv 1777

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

Hypothesis

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

Assertion

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

Distinct variable group:   ph,x

Proof of Theorem reubidv

Step	Hyp	Ref	Expression
1		reubidv.1	. . 3 |- (ph -> (ps <-> ch))
2	1	adantr 389	. 2 |- ((ph /\ x e. A) -> (ps <-> ch))
3	2	reubidva 1776	1 |- (ph -> (E!x e. A ps <-> E!x e. A ch))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 956  E!wreu 1644

This theorem is referenced by:  reueqd 1790  oawordeu 4179  aceq6b 4722  riesz4t 9935  cnlnadjlem4 9941  cnlnadjeut 9949

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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