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Theorem nnral 2447
Description: The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1629. (Contributed by Jim Kingdon, 1-Aug-2024.)
Assertion
Ref Expression
nnral (¬ ¬ ∀𝑥𝐴 𝜑 → ∀𝑥𝐴 ¬ ¬ 𝜑)

Proof of Theorem nnral
StepHypRef Expression
1 rexnalim 2446 . . 3 (∃𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 𝜑)
21con3i 622 . 2 (¬ ¬ ∀𝑥𝐴 𝜑 → ¬ ∃𝑥𝐴 ¬ 𝜑)
3 ralnex 2445 . 2 (∀𝑥𝐴 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)
42, 3sylibr 133 1 (¬ ¬ ∀𝑥𝐴 𝜑 → ∀𝑥𝐴 ¬ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  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-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514
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:  onntri13  7174  onntri24  7178
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