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Theorem nnal 1642
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
nnal (¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)

Proof of Theorem nnal
StepHypRef Expression
1 exnalim 1639 . . 3 (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
21con3i 627 . 2 (¬ ¬ ∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
3 alnex 1492 . 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 1346  wex 1485
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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454
This theorem is referenced by:  bj-stal  13784
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