Theorem nrexdv 1271

Description: Deduction adding restricted existential quantifier to negated wff.

Hypothesis

Ref	Expression
nrexdv.1	|- ((ph /\ x e. A) -> -. ps)

Assertion

Ref	Expression
nrexdv	|- (ph -> -. E.x e. A ps)

Distinct variable group(s):   ph,x

Proof of Theorem nrexdv

Step	Hyp	Ref	Expression
1		nrexdv.1	. . . 4 |- ((ph /\ x e. A) -> -. ps)
2	1	exp 291	. . 3 |- (ph -> (x e. A -> -. ps))
3	2	r19.21aiv 1259	. 2 |- (ph -> A.x e. A -. ps)
4		ralnex 1209	. 2 |- (A.x e. A -. ps <-> -. E.x e. A ps)
5	3, 4	sylib 173	1 |- (ph -> -. E.x e. A ps)

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   e. wcel 1092  A.wral 1201  E.wrex 1202

This theorem is referenced by:  class2set 1747  peano5 2394  oalimcl 3162  setind 3492  cardlim 3657  cardaleph 3690

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-ral 1205  df-rex 1206

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