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Theorem bj-nnal 13033
 Description: The double negation of a universal quantification implies the universal quantification of the double negation. (Contributed by BJ, 24-Nov-2023.)
Assertion
Ref Expression
bj-nnal (¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)

Proof of Theorem bj-nnal
StepHypRef Expression
1 exnalim 1625 . . 3 (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
21con3i 621 . 2 (¬ ¬ ∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
3 alnex 1475 . 2 (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
42, 3sylibr 133 1 (¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1329  ∃wex 1468 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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437 This theorem is referenced by:  bj-stal  13041
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