Theorem reubidv 2681

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 17-Oct-1996.)

Hypothesis

Ref	Expression
reubidv.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
reubidv	|- ( 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 reubidv

Step	Hyp	Ref	Expression
1		reubidv.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	adantr 276	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	reubidva 2680	1 |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2167   E!wreu 2477

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-eu 2048  df-reu 2482

This theorem is referenced by:  reueqd  2707  sbcreug  3070  xpf1o  6905  srpospr  7850  creur  8986  creui  8987  divalg2  12091  srgideu  13528  ringideu  13573