Theorem 3ori

Description: Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006)

	Ref	Expression
Hypothesis	3ori.1	$$⊢\left(\phi \vee \psi \vee \chi \right)$$
Assertion	3ori	$$⊢\left(\neg \phi \wedge \neg \psi \right)\to \chi$$

Proof

Step	Hyp	Ref	Expression
1		3ori.1	$$⊢\left(\phi \vee \psi \vee \chi \right)$$
2		ioran	$$⊢\neg \left(\phi \vee \psi \right)\leftrightarrow \left(\neg \phi \wedge \neg \psi \right)$$
3		df-3or	$$⊢\left(\phi \vee \psi \vee \chi \right)\leftrightarrow \left(\left(\phi \vee \psi \right)\vee \chi \right)$$
4	1 3	mpbi	$$⊢\left(\left(\phi \vee \psi \right)\vee \chi \right)$$
5	4	ori	$$⊢\neg \left(\phi \vee \psi \right)\to \chi$$
6	2 5	sylbir	$$⊢\left(\neg \phi \wedge \neg \psi \right)\to \chi$$