Theorem nbn

Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 21-Jun-1993) (Proof shortened by Wolf Lammen, 3-Oct-2013)

	Ref	Expression
Hypothesis	nbn.1	⊢ ¬ 𝜑
Assertion	nbn	⊢ ( ¬ 𝜓 ↔ ( 𝜓 ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nbn.1	⊢ ¬ 𝜑
2		bibif	⊢ ( ¬ 𝜑 → ( ( 𝜓 ↔ 𝜑 ) ↔ ¬ 𝜓 ) )
3	1 2	ax-mp	⊢ ( ( 𝜓 ↔ 𝜑 ) ↔ ¬ 𝜓 )
4	3	bicomi	⊢ ( ¬ 𝜓 ↔ ( 𝜓 ↔ 𝜑 ) )