Theorem nrexdv 2412

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 2392	. 2 |- ( ph -> A. x e. A -. ps )
3		ralnex 2316	. 2 |- ( A. x e. A -. ps <-> -. E. x e. A ps )
4	2, 3	sylib 127	1 |- ( ph -> -. E. x e. A ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   e. wcel 1393   A.wral 2306   E.wrex 2307

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-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-4 1400  ax-17 1419

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-ral 2311  df-rex 2312

This theorem is referenced by:  ltpopr  6693  cauappcvgprlemladdru  6754  cauappcvgprlemladdrl  6755  caucvgprlemladdrl  6776  caucvgprprlemaddq  6806