Theorem bibif

Description: Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007) (Proof shortened by Wolf Lammen, 28-Jan-2013)

	Ref	Expression
Assertion	bibif	⊢ ( ¬ 𝜓 → ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nbn2	⊢ ( ¬ 𝜓 → ( ¬ 𝜑 ↔ ( 𝜓 ↔ 𝜑 ) ) )
2		bicom	⊢ ( ( 𝜓 ↔ 𝜑 ) ↔ ( 𝜑 ↔ 𝜓 ) )
3	1 2	syl6rbb	⊢ ( ¬ 𝜓 → ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ 𝜑 ) )