Theorem reubidv 2737

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 2736	1 |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   e. wcel 2209   E!wreu 2530

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-nf 1514  df-eu 2089  df-reu 2535

This theorem is used by:  reueqd  2763  sbcreug  3132  xpf1o  7144  srpospr  8150  creur  9289  creui  9290  divalg2  12693  srgideu  14276  ringideu  14321