Theorem con2bi

Description: Contraposition. Theorem *4.12 of WhiteheadRussell p. 117. (Contributed by NM, 15-Apr-1995) (Proof shortened by Wolf Lammen, 3-Jan-2013)

	Ref	Expression
Assertion	con2bi	⊢ ( ( 𝜑 ↔ ¬ 𝜓 ) ↔ ( 𝜓 ↔ ¬ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		notbi	⊢ ( ( 𝜑 ↔ ¬ 𝜓 ) ↔ ( ¬ 𝜑 ↔ ¬ ¬ 𝜓 ) )
2		notnotb	⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 )
3	2	bibi2i	⊢ ( ( ¬ 𝜑 ↔ 𝜓 ) ↔ ( ¬ 𝜑 ↔ ¬ ¬ 𝜓 ) )
4		bicom	⊢ ( ( ¬ 𝜑 ↔ 𝜓 ) ↔ ( 𝜓 ↔ ¬ 𝜑 ) )
5	1 3 4	3bitr2i	⊢ ( ( 𝜑 ↔ ¬ 𝜓 ) ↔ ( 𝜓 ↔ ¬ 𝜑 ) )