Theorem nrex 2589

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 2550	. 2 |- A. x e. A -. ps
3		ralnex 2485	. 2 |- ( A. x e. A -. ps <-> -. E. x e. A ps )
4	2, 3	mpbi 145	1 |- -. E. x e. A ps

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2167   A.wral 2475   E.wrex 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie2 1508

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-ral 2480  df-rex 2481

This theorem is referenced by:  rex0  3468  iun0  3973  canth  5875  frec0g  6455  nominpos  9229  sqrt2irr  12330  gsum0g  13039  exmidsbthrlem  15666