Theorem nrex 2454

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 2417	. 2 |- A. x e. A -. ps
3		ralnex 2359	. 2 |- ( A. x e. A -. ps <-> -. E. x e. A ps )
4	2, 3	mpbi 143	1 |- -. E. x e. A ps

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1434   A.wral 2349   E.wrex 2350

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 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie2 1424

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-ral 2354  df-rex 2355

This theorem is referenced by:  rex0  3272  iun0  3742  frec0g  6046  nominpos  8335  sqrt2irr  10685