Theorem notbi

Description: Contraposition. Theorem *4.11 of WhiteheadRussell p. 117. (Contributed by NM, 21-May-1994) (Proof shortened by Wolf Lammen, 12-Jun-2013)

	Ref	Expression
Assertion	notbi	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ¬ 𝜑 ↔ ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 ↔ 𝜓 ) )
2	1	notbid	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
3		id	⊢ ( ( ¬ 𝜑 ↔ ¬ 𝜓 ) → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
4	3	con4bid	⊢ ( ( ¬ 𝜑 ↔ ¬ 𝜓 ) → ( 𝜑 ↔ 𝜓 ) )
5	2 4	impbii	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ¬ 𝜑 ↔ ¬ 𝜓 ) )