Theorem nrexdv 2454

Description: Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)

Hypothesis

Ref	Expression
nrexdv.1	|- ( ( ph /\ x e. A ) -> -. ps )

Assertion

Ref	Expression
nrexdv	|- ( ph -> -. E. x e. A ps )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem nrexdv

Step	Hyp	Ref	Expression
1		nrexdv.1	. . 3 |- ( ( ph /\ x e. A ) -> -. ps )
2	1	ralrimiva 2434	. 2 |- ( ph -> A. x e. A -. ps )
3		ralnex 2358	. 2 |- ( A. x e. A -. ps <-> -. E. x e. A ps )
4	2, 3	sylib 120	1 |- ( ph -> -. E. x e. A ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   e. wcel 1433   A.wral 2348   E.wrex 2349

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-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie2 1423  ax-4 1440  ax-17 1459

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-ral 2353  df-rex 2354

This theorem is referenced by:  ltpopr  6785  cauappcvgprlemladdru  6846  cauappcvgprlemladdrl  6847  caucvgprlemladdrl  6868  caucvgprprlemaddq  6898  dvdsle  10244