Theorem rexbidv2 2497

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 1836	. 2 |- ( ph -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. B /\ ch ) ) )
3		df-rex 2478	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
4		df-rex 2478	. 2 |- ( E. x e. B ch <-> E. x ( x e. B /\ ch ) )
5	2, 3, 4	3bitr4g 223	1 |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1503   e. wcel 2164   E.wrex 2473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-rex 2478

This theorem is referenced by:  rexss  3246  rexsupp  5682  isoini  5861  elfi2  7031  ltexpi  7397  rexuz  9645  4sqexercise2  12537  4sqlemsdc  12538  sscoll2  15480