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)

	Ref	Expression
Assertion	nbbn	$$⊢\left(\neg \phi \leftrightarrow \psi \right)\leftrightarrow \neg \left(\phi \leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		xor3	$$⊢\neg \left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\phi \leftrightarrow \neg \psi \right)$$
2		con2bi	$$⊢\left(\phi \leftrightarrow \neg \psi \right)\leftrightarrow \left(\psi \leftrightarrow \neg \phi \right)$$
3		bicom	$$⊢\left(\psi \leftrightarrow \neg \phi \right)\leftrightarrow \left(\neg \phi \leftrightarrow \psi \right)$$
4	1 2 3	3bitrri	$$⊢\left(\neg \phi \leftrightarrow \psi \right)\leftrightarrow \neg \left(\phi \leftrightarrow \psi \right)$$