Theorem birexdv2 1222

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

Hypothesis

Ref	Expression
birexdv2.1	|- (ph -> ((x e. A /\ ps) <-> (x e. B /\ ch)))

Assertion

Ref	Expression
birexdv2	|- (ph -> (E.x e. A ps <-> E.x e. B ch))

Distinct variable group(s):   ph,x

Proof of Theorem birexdv2

Step	Hyp	Ref	Expression
1		birexdv2.1	. . 3 |- (ph -> ((x e. A /\ ps) <-> (x e. B /\ ch)))
2	1	biexdv 936	. 2 |- (ph -> (E.x(x e. A /\ ps) <-> E.x(x e. B /\ ch)))
3		df-rex 1206	. 2 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
4		df-rex 1206	. 2 |- (E.x e. B ch <-> E.x(x e. B /\ ch))
5	2, 3, 4	3bitr4g 428	1 |- (ph -> (E.x e. A ps <-> E.x e. B ch))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  isoini 2938  nnaordex 3191  nnawordex 3192

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  ax-17 925

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

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