Theorem nrex 1729

Description: Inference adding restricted existential quantifier to negated wff.

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 1698	. 2 |- A.x e. A -. ps
3		ralnex 1653	. 2 |- (A.x e. A -. ps <-> -. E.x e. A ps)
4	2, 3	mpbi 189	1 |- -. E.x e. A ps

Syntax hints:  -. wn 2   -> wi 3   e. wcel 958  A.wral 1645  E.wrex 1646

This theorem is referenced by:  rex0 2291  iun0 2604  orduninsuc 3114  cfsuc 4915  nominpos 6043  nnunb 6070  indstr 6461  sqr2irrlem3 6726  climubi 7153  eirr 7394  ruclem37 7546  hatomistic 10289

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-ral 1649  df-rex 1650

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