Theorem nrex 2716

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 2679	. 2 |- A.x e. A -. ps
3		ralnex 2624	. 2 |- (A.x e. A -. ps <-> -. E.x e. A ps)
4	2, 3	mpbi 199	1 |- -. E.x e. A ps

Syntax hints:  -. wn 3   -> wi 4   e. wcel 1710  A.wral 2614  E.wrex 2615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-ral 2619  df-rex 2620

This theorem is referenced by:  rex0  3563  iun0  4022  0nelsuc  4400  addcnul1  4452  nulnnn  4556  0cnelphi  4597  proj1op  4600  proj2op  4601  nenpw1pwlem2  6085  nchoice  6308  fnfreclem2  6318