Theorem 3ioran

Description: Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011)

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

Proof

Step	Hyp	Ref	Expression
1		ioran	$$⊢\neg \left(\phi \vee \psi \right)\leftrightarrow \left(\neg \phi \wedge \neg \psi \right)$$
2	1	anbi1i	$$⊢\left(\neg \left(\phi \vee \psi \right)\wedge \neg \chi \right)\leftrightarrow \left(\left(\neg \phi \wedge \neg \psi \right)\wedge \neg \chi \right)$$
3		ioran	$$⊢\neg \left(\left(\phi \vee \psi \right)\vee \chi \right)\leftrightarrow \left(\neg \left(\phi \vee \psi \right)\wedge \neg \chi \right)$$
4		df-3or	$$⊢\left(\phi \vee \psi \vee \chi \right)\leftrightarrow \left(\left(\phi \vee \psi \right)\vee \chi \right)$$
5	3 4	xchnxbir	$$⊢\neg \left(\phi \vee \psi \vee \chi \right)\leftrightarrow \left(\neg \left(\phi \vee \psi \right)\wedge \neg \chi \right)$$
6		df-3an	$$⊢\left(\neg \phi \wedge \neg \psi \wedge \neg \chi \right)\leftrightarrow \left(\left(\neg \phi \wedge \neg \psi \right)\wedge \neg \chi \right)$$
7	2 5 6	3bitr4i	$$⊢\neg \left(\phi \vee \psi \vee \chi \right)\leftrightarrow \left(\neg \phi \wedge \neg \psi \wedge \neg \chi \right)$$