Theorem nnal 1695

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 1692	. . 3 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
2	1	con3i 635	. 2 ⊢ (¬ ¬ ∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
3		alnex 1545	. 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
4	2, 3	sylibr 134	1 ⊢ (¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1393  ∃wex 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507

This theorem is referenced by:  bj-stal  16071