Theorem nbn

Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 21-Jun-1993) (Proof shortened by Wolf Lammen, 3-Oct-2013)

	Ref	Expression
Hypothesis	nbn.1	$$⊢\neg \phi$$
Assertion	nbn	$$⊢\neg \psi \leftrightarrow \left(\psi \leftrightarrow \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nbn.1	$$⊢\neg \phi$$
2		bibif	$$⊢\neg \phi \to \left(\left(\psi \leftrightarrow \phi \right)\leftrightarrow \neg \psi \right)$$
3	1 2	ax-mp	$$⊢\left(\psi \leftrightarrow \phi \right)\leftrightarrow \neg \psi$$
4	3	bicomi	$$⊢\neg \psi \leftrightarrow \left(\psi \leftrightarrow \phi \right)$$