Theorem rexbidv2 2469

Description: Formula-building rule for restricted existential quantifier (deduction form). (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 1813	. 2 |- ( ph -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. B /\ ch ) ) )
3		df-rex 2450	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
4		df-rex 2450	. 2 |- ( E. x e. B ch <-> E. x ( x e. B /\ ch ) )
5	2, 3, 4	3bitr4g 222	1 |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   E.wex 1480   e. wcel 2136   E.wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-rex 2450

This theorem is referenced by:  rexss  3209  rexsupp  5609  isoini  5786  elfi2  6937  ltexpi  7278  rexuz  9518  sscoll2  13870