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Theorem nrexdv 2570
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 2550 . 2 (𝜑 → ∀𝑥𝐴 ¬ 𝜓)
3 ralnex 2465 . 2 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
42, 3sylib 122 1 (𝜑 → ¬ ∃𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wcel 2148  wral 2455  wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie2 1494  ax-4 1510  ax-17 1526
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-ral 2460  df-rex 2461
This theorem is referenced by:  ltpopr  7572  cauappcvgprlemladdru  7633  cauappcvgprlemladdrl  7634  caucvgprlemladdrl  7655  caucvgprprlemaddq  7685  dvdsle  11820
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