Theorem nnnotnotr 16311

Description: Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 855, 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 657	. 2 ⊢ (¬ (¬ ¬ 𝜑 → 𝜑) → ¬ 𝜑)
2		pm2.24 624	. . 3 ⊢ (¬ 𝜑 → (¬ ¬ 𝜑 → 𝜑))
3	2	con3i 635	. 2 ⊢ (¬ (¬ ¬ 𝜑 → 𝜑) → ¬ ¬ 𝜑)
4	1, 3	pm2.65i 642	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 617  ax-in2 618

This theorem is referenced by: (None)