Theorem reubidv 2493

Description: Formula-building rule for restricted existential quantifier (deduction rule). (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 261	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	reubidva 2492	1 |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1393   E!wreu 2308

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-eu 1903  df-reu 2313

This theorem is referenced by:  reueqd  2515  sbcreug  2838  srpospr  6867  creur  7911  creui  7912