Theorem notbii

Description: Negate both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993) (Proof shortened by Wolf Lammen, 19-May-2013)

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

Proof

Step	Hyp	Ref	Expression
1		notbii.1	⊢ ( 𝜑 ↔ 𝜓 )
2		notbi	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
3	1 2	mpbi	⊢ ( ¬ 𝜑 ↔ ¬ 𝜓 )