Theorem reubidv 2550

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 270	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	reubidva 2549	1 |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1438   E!wreu 2361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-eu 1951  df-reu 2366

This theorem is referenced by:  reueqd  2572  sbcreug  2919  xpf1o  6560  srpospr  7328  creur  8419  creui  8420  divalg2  11204