Theorem birex2 1227

Description: Inference adding different restricted existential quantifiers to each side of an equivalence.

Hypothesis

Ref	Expression
birex2.1	|- ((x e. A /\ ph) <-> (x e. B /\ ps))

Assertion

Ref	Expression
birex2	|- (E.x e. A ph <-> E.x e. B ps)

Proof of Theorem birex2

Step	Hyp	Ref	Expression
1		birex2.1	. . 3 |- ((x e. A /\ ph) <-> (x e. B /\ ps))
2	1	biex 733	. 2 |- (E.x(x e. A /\ ph) <-> E.x(x e. B /\ ps))
3		df-rex 1206	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4		df-rex 1206	. 2 |- (E.x e. B ps <-> E.x(x e. B /\ ps))
5	2, 3, 4	3bitr4 158	1 |- (E.x e. A ph <-> E.x e. B ps)

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  birexa 1229  wefrc 2195  bnd2 3549  sumdmdi 5785

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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