Theorem nbbn

Description: Move negation outside of biconditional. Compare Theorem *5.18 of WhiteheadRussell p. 124. (Contributed by NM, 27-Jun-2002) (Proof shortened by Wolf Lammen, 20-Sep-2013) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026)

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

Proof

Step	Hyp	Ref	Expression
1		pm5.18	⊢ ( ( ¬ 𝜑 ↔ 𝜓 ) ↔ ¬ ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
2		notbi	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
3	1 2	xchbinxr	⊢ ( ( ¬ 𝜑 ↔ 𝜓 ) ↔ ¬ ( 𝜑 ↔ 𝜓 ) )