Theorem notbid

Description: Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994)

	Ref	Expression
Hypothesis	notbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	notbid	⊢ ( 𝜑 → ( ¬ 𝜓 ↔ ¬ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		notbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		notnotb	⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 )
3		notnotb	⊢ ( 𝜒 ↔ ¬ ¬ 𝜒 )
4	1 2 3	3bitr3g	⊢ ( 𝜑 → ( ¬ ¬ 𝜓 ↔ ¬ ¬ 𝜒 ) )
5	4	con4bid	⊢ ( 𝜑 → ( ¬ 𝜓 ↔ ¬ 𝜒 ) )