Theorem notnotnot 635

Description: Triple negation is equivalent to negation. (Contributed by Jim Kingdon, 28-Jul-2018.)

Assertion

Ref	Expression
notnotnot	⊢ (¬ ¬ ¬ 𝜑 ↔ ¬ 𝜑)

Proof of Theorem notnotnot

Step	Hyp	Ref	Expression
1		notnot 630	. . 3 ⊢ (𝜑 → ¬ ¬ 𝜑)
2	1	con3i 633	. 2 ⊢ (¬ ¬ ¬ 𝜑 → ¬ 𝜑)
3		notnot 630	. 2 ⊢ (¬ 𝜑 → ¬ ¬ ¬ 𝜑)
4	2, 3	impbii 126	1 ⊢ (¬ ¬ ¬ 𝜑 ↔ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  stabnot  834  dcnnOLD  850  bj-nnsn  15379