Theorem nrex 2717

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 2680	. 2 |- A.x e. A -. ps
3		ralnex 2625	. 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 2615  E.wrex 2616

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-ral 2620  df-rex 2621

This theorem is referenced by:  rex0  3564  iun0  4023  0nelsuc  4401  addcnul1  4453  nulnnn  4557  0cnelphi  4598  proj1op  4601  proj2op  4602  nenpw1pwlem2  6086  nchoice  6309  fnfreclem2  6319