Theorem rexbidv2 2371

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 1746	. 2 |- ( ph -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. B /\ ch ) ) )
3		df-rex 2354	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
4		df-rex 2354	. 2 |- ( E. x e. B ch <-> E. x ( x e. B /\ ch ) )
5	2, 3, 4	3bitr4g 221	1 |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   E.wex 1421   e. wcel 1433   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-rex 2354

This theorem is referenced by:  rexss  3061  rexsupp  5312  isoini  5477  ltexpi  6527  rexuz  8668