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Theorem nrexdv 2717
 Description: Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
Hypothesis
Ref Expression
nrexdv.1 ((φ x A) → ¬ ψ)
Assertion
Ref Expression
nrexdv (φ → ¬ x A ψ)
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   A(x)

Proof of Theorem nrexdv
StepHypRef Expression
1 nrexdv.1 . . 3 ((φ x A) → ¬ ψ)
21ralrimiva 2697 . 2 (φx A ¬ ψ)
3 ralnex 2624 . 2 (x A ¬ ψ ↔ ¬ x A ψ)
42, 3sylib 188 1 (φ → ¬ x A ψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∈ wcel 1710  ∀wral 2614  ∃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
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