Theorem 3anor

Description: Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009) (Proof shortened by Wolf Lammen, 8-Apr-2022)

	Ref	Expression
Assertion	3anor	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \neg \left(\neg \phi \vee \neg \psi \vee \neg \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		3ianor	$$⊢\neg \left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\neg \phi \vee \neg \psi \vee \neg \chi \right)$$
2	1	con1bii	$$⊢\neg \left(\neg \phi \vee \neg \psi \vee \neg \chi \right)\leftrightarrow \left(\phi \wedge \psi \wedge \chi \right)$$
3	2	bicomi	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \neg \left(\neg \phi \vee \neg \psi \vee \neg \chi \right)$$