Theorem 3ianor

Description: Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009) (Proof shortened by Andrew Salmon, 13-May-2011) (Revised by Wolf Lammen, 8-Apr-2022)

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

Proof

Step	Hyp	Ref	Expression
1		ianor	$$⊢\neg \left(\phi \wedge \psi \right)\leftrightarrow \left(\neg \phi \vee \neg \psi \right)$$
2	1	orbi1i	$$⊢\left(\neg \left(\phi \wedge \psi \right)\vee \neg \chi \right)\leftrightarrow \left(\left(\neg \phi \vee \neg \psi \right)\vee \neg \chi \right)$$
3		ianor	$$⊢\neg \left(\left(\phi \wedge \psi \right)\wedge \chi \right)\leftrightarrow \left(\neg \left(\phi \wedge \psi \right)\vee \neg \chi \right)$$
4		df-3an	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\wedge \chi \right)$$
5	3 4	xchnxbir	$$⊢\neg \left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\neg \left(\phi \wedge \psi \right)\vee \neg \chi \right)$$
6		df-3or	$$⊢\left(\neg \phi \vee \neg \psi \vee \neg \chi \right)\leftrightarrow \left(\left(\neg \phi \vee \neg \psi \right)\vee \neg \chi \right)$$
7	2 5 6	3bitr4i	$$⊢\neg \left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\neg \phi \vee \neg \psi \vee \neg \chi \right)$$