Theorem nnal 1702

Description: The double negation of a universal quantification implies the universal quantification of the double negation. The converse holds in classical but not in intuitionistic logic. (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
nnal	⊢ (¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)

Proof of Theorem nnal

Step	Hyp	Ref	Expression
1		exnalim 1699	. . 3 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
2	1	con3i 641	. 2 ⊢ (¬ ¬ ∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
3		alnex 1552	. 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
4	2, 3	sylibr 134	1 ⊢ (¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1400  ∃wex 1545

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 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514

This theorem is referenced by:  bj-stal  16691