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Theorem nrexdv 2502
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 2482 . 2 (𝜑 → ∀𝑥𝐴 ¬ 𝜓)
3 ralnex 2403 . 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 1465  wral 2393  wrex 2394
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 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-ie2 1455  ax-4 1472  ax-17 1491
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-ral 2398  df-rex 2399
This theorem is referenced by:  ltpopr  7371  cauappcvgprlemladdru  7432  cauappcvgprlemladdrl  7433  caucvgprlemladdrl  7454  caucvgprprlemaddq  7484  dvdsle  11469
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