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Theorem nrexdv 2412
 Description: Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
Hypothesis
Ref Expression
nrexdv.1 ((𝜑𝑥𝐴) → ¬ 𝜓)
Assertion
Ref Expression
nrexdv (𝜑 → ¬ ∃𝑥𝐴 𝜓)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem nrexdv
StepHypRef Expression
1 nrexdv.1 . . 3 ((𝜑𝑥𝐴) → ¬ 𝜓)
21ralrimiva 2392 . 2 (𝜑 → ∀𝑥𝐴 ¬ 𝜓)
3 ralnex 2316 . 2 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
42, 3sylib 127 1 (𝜑 → ¬ ∃𝑥𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-4 1400  ax-17 1419 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-ral 2311  df-rex 2312 This theorem is referenced by:  ltpopr  6693  cauappcvgprlemladdru  6754  cauappcvgprlemladdrl  6755  caucvgprlemladdrl  6776  caucvgprprlemaddq  6806
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