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

Step	Hyp	Ref	Expression
1		exnalim 1639	. . 3 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
2	1	con3i 627	. 2 ⊢ (¬ ¬ ∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
3		alnex 1492	. 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
4	2, 3	sylibr 133	1 ⊢ (¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)

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