Theorem reubidva 2689

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 1551	. 2 |- F/ x ph
2		reubidva.1	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	1, 2	reubida 2688	1 |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2176   E!wreu 2486

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-eu 2057  df-reu 2491

This theorem is referenced by:  reubidv  2690  f1ofveu  5932  srpospr  7896  icoshftf1o  10113  divalgb  12236  1arith2  12691