Theorem nnnotnotr 14025

Description: Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 845, it can be proved with reference only to implication and negation (that is, without use of disjunction). (Contributed by Jim Kingdon, 21-Oct-2024.)

Assertion

Ref	Expression
nnnotnotr	⊢ ¬ ¬ (¬ ¬ 𝜑 → 𝜑)

Proof of Theorem nnnotnotr

Step	Hyp	Ref	Expression
1		conax1 648	. 2 ⊢ (¬ (¬ ¬ 𝜑 → 𝜑) → ¬ 𝜑)
2		pm2.24 616	. . 3 ⊢ (¬ 𝜑 → (¬ ¬ 𝜑 → 𝜑))
3	2	con3i 627	. 2 ⊢ (¬ (¬ ¬ 𝜑 → 𝜑) → ¬ ¬ 𝜑)
4	1, 3	pm2.65i 634	1 ⊢ ¬ ¬ (¬ ¬ 𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 609  ax-in2 610

This theorem is referenced by: (None)