Theorem nannot

Description: Negation in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007) (Revised by Wolf Lammen, 26-Jun-2020)

	Ref	Expression
Assertion	nannot	⊢ ( ¬ 𝜑 ↔ ( 𝜑 ⊼ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		dfnan2	⊢ ( ( 𝜑 ⊼ 𝜑 ) ↔ ( 𝜑 → ¬ 𝜑 ) )
2		pm4.8	⊢ ( ( 𝜑 → ¬ 𝜑 ) ↔ ¬ 𝜑 )
3	1 2	bitr2i	⊢ ( ¬ 𝜑 ↔ ( 𝜑 ⊼ 𝜑 ) )