Theorem birexa 1229

Description: Inference adding restricted existential quantifier to both sides of an equivalence.

Hypothesis

Ref	Expression
birala.1	|- (x e. A -> (ph <-> ps))

Assertion

Ref	Expression
birexa	|- (E.x e. A ph <-> E.x e. A ps)

Proof of Theorem birexa

Step	Hyp	Ref	Expression
1		birala.1	. . 3 |- (x e. A -> (ph <-> ps))
2	1	pm5.32i 489	. 2 |- ((x e. A /\ ph) <-> (x e. A /\ ps))
3	2	birex2 1227	1 |- (E.x e. A ph <-> E.x e. A ps)

Syntax hints:   -> wi 2   <-> wb 127   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  bi2rexa 1230  elrnopab 2884  f1oweOLD 2944  unbndrank 3527  pjpj0 5259  atom1d 5750

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

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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