Theorem nrexdv 2717

Description: Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)

Hypothesis

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

Assertion

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

Distinct variable group:   ph,x

Allowed substitution hints:   ps(x)   A(x)

Proof of Theorem nrexdv

Step	Hyp	Ref	Expression
1		nrexdv.1	. . 3 |- ((ph /\ x e. A) -> -. ps)
2	1	ralrimiva 2697	. 2 |- (ph -> A.x e. A -. ps)
3		ralnex 2624	. 2 |- (A.x e. A -. ps <-> -. E.x e. A ps)
4	2, 3	sylib 188	1 |- (ph -> -. E.x e. A ps)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   e. wcel 1710  A.wral 2614  E.wrex 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2619  df-rex 2620

This theorem is referenced by:  nndisjeq  4429  ltfinirr  4457  nnadjoin  4520  tfinnn  4534  vfinncvntsp  4549