Theorem reubidva 2652

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004.)

Hypothesis

Ref	Expression
reubidva.1	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
reubidva	|- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)

Proof of Theorem reubidva

Step	Hyp	Ref	Expression
1		nfv 1521	. 2 |- F/ x ph
2		reubidva.1	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	1, 2	reubida 2651	1 |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2141   E!wreu 2450

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-eu 2022  df-reu 2455

This theorem is referenced by:  reubidv  2653  f1ofveu  5841  srpospr  7745  icoshftf1o  9948  divalgb  11884  1arith2  12320