Theorem nrex 2411

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

Hypothesis

Ref	Expression
nrex.1	|- ( x e. A -> -. ps )

Assertion

Ref	Expression
nrex	|- -. E. x e. A ps

Proof of Theorem nrex

Step	Hyp	Ref	Expression
1		nrex.1	. . 3 |- ( x e. A -> -. ps )
2	1	rgen 2374	. 2 |- A. x e. A -. ps
3		ralnex 2316	. 2 |- ( A. x e. A -. ps <-> -. E. x e. A ps )
4	2, 3	mpbi 133	1 |- -. E. x e. A ps

Syntax hints:   -. wn 3   -> wi 4   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

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

This theorem is referenced by:  rex0  3238  iun0  3713  frec0g  5983  nominpos  8162  sqrt2irr  9878