Theorem nrex 2558

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 2519	. 2 |- A. x e. A -. ps
3		ralnex 2454	. 2 |- ( A. x e. A -. ps <-> -. E. x e. A ps )
4	2, 3	mpbi 144	1 |- -. E. x e. A ps

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2136   A.wral 2444   E.wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie2 1482

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-ral 2449  df-rex 2450

This theorem is referenced by:  rex0  3426  iun0  3922  canth  5796  frec0g  6365  nominpos  9094  sqrt2irr  12094  exmidsbthrlem  13901