Theorem rexbidv2 2638

Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 22-May-1999.)

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

Allowed substitution hints:   ps(x)   ch(x)   A(x)   B(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 1626	. 2 |- (ph -> (E.x(x e. A /\ ps) <-> E.x(x e. B /\ ch)))
3		df-rex 2621	. 2 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
4		df-rex 2621	. 2 |- (E.x e. B ch <-> E.x(x e. B /\ ch))
5	2, 3, 4	3bitr4g 279	1 |- (ph -> (E.x e. A ps <-> E.x e. B ch))

This proof depends on syntax axioms:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   e. wcel 1710  E.wrex 2616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616

This proof depends on definitions:  df-bi 177  df-ex 1542  df-rex 2621

This theorem is used by:  rexss  3334  isoini  5498