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	$$⊢\left(\neg \phi \leftrightarrow \psi \right)\leftrightarrow \neg \left(\phi \leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm5.18	$$⊢\left(\neg \phi \leftrightarrow \psi \right)\leftrightarrow \neg \left(\neg \phi \leftrightarrow \neg \psi \right)$$
2		notbi	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\neg \phi \leftrightarrow \neg \psi \right)$$
3	1 2	xchbinxr	$$⊢\left(\neg \phi \leftrightarrow \psi \right)\leftrightarrow \neg \left(\phi \leftrightarrow \psi \right)$$