Theorem rexbidv2 1666

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

Hypothesis

Ref	Expression
rexbidv2.1	|- (ph -> ((x e. A /\ ps) <-> (x e. B /\ ch)))

Assertion

Ref	Expression
rexbidv2	|- (ph -> (E.x e. A ps <-> E.x e. B ch))

Distinct variable group:   ph,x

Proof of Theorem rexbidv2

Step	Hyp	Ref	Expression
1		rexbidv2.1	. . 3 |- (ph -> ((x e. A /\ ps) <-> (x e. B /\ ch)))
2	1	exbidv 1279	. 2 |- (ph -> (E.x(x e. A /\ ps) <-> E.x(x e. B /\ ch)))
3		df-rex 1650	. 2 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
4		df-rex 1650	. 2 |- (E.x e. B ch <-> E.x(x e. B /\ ch))
5	2, 3, 4	3bitr4g 555	1 |- (ph -> (E.x e. A ps <-> E.x e. B ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 958  E.wex 980  E.wrex 1646

This theorem is referenced by:  isoini 3900  nnaordex 4249  nnawordex 4250  rexuz 6444

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-rex 1650

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