Theorem reubidv 2678

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 2677	1 |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2164   E!wreu 2474

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-eu 2045  df-reu 2479

This theorem is referenced by:  reueqd  2704  sbcreug  3066  xpf1o  6900  srpospr  7843  creur  8978  creui  8979  divalg2  12067  srgideu  13468  ringideu  13513