Theorem nrex 2562

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 2523	. 2 |- A. x e. A -. ps
3		ralnex 2458	. 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 2141   A.wral 2448   E.wrex 2449

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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie2 1487

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-ral 2453  df-rex 2454

This theorem is referenced by:  rex0  3432  iun0  3929  canth  5807  frec0g  6376  nominpos  9115  sqrt2irr  12116  exmidsbthrlem  14054