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Theorem nrexdv 2429
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 2409 . 2 (𝜑 → ∀𝑥𝐴 ¬ 𝜓)
3 ralnex 2333 . 2 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
42, 3sylib 131 1 (𝜑 → ¬ ∃𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wcel 1409  wral 2323  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie2 1399  ax-4 1416  ax-17 1435
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-ral 2328  df-rex 2329
This theorem is referenced by:  ltpopr  6751  cauappcvgprlemladdru  6812  cauappcvgprlemladdrl  6813  caucvgprlemladdrl  6834  caucvgprprlemaddq  6864  dvdsle  10156
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