Theorem nrex 2527

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 2488	. 2 |- A. x e. A -. ps
3		ralnex 2427	. 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 1481   A.wral 2417   E.wrex 2418

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 1424  ax-gen 1426  ax-ie2 1471

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-ral 2422  df-rex 2423

This theorem is referenced by:  rex0  3385  iun0  3877  frec0g  6302  nominpos  8981  sqrt2irr  11876  exmidsbthrlem  13392