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Theorem nrexdv 2550
 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 2530 . 2 (𝜑 → ∀𝑥𝐴 ¬ 𝜓)
3 ralnex 2445 . 2 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
42, 3sylib 121 1 (𝜑 → ¬ ∃𝑥𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1427  ax-gen 1429  ax-ie2 1474  ax-4 1490  ax-17 1506 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-ral 2440  df-rex 2441 This theorem is referenced by:  ltpopr  7498  cauappcvgprlemladdru  7559  cauappcvgprlemladdrl  7560  caucvgprlemladdrl  7581  caucvgprprlemaddq  7611  dvdsle  11717
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