Theorem rexbidv2 2329

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 1706	. 2 |- ( ph -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. B /\ ch ) ) )
3		df-rex 2312	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
4		df-rex 2312	. 2 |- ( E. x e. B ch <-> E. x ( x e. B /\ ch ) )
5	2, 3, 4	3bitr4g 212	1 |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   E.wex 1381   e. wcel 1393   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-rex 2312

This theorem is referenced by:  rexss  3007  rexsupp  5291  isoini  5457  ltexpi  6435  rexuz  8523