Theorem 3orel1

Description: Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011)

	Ref	Expression
Assertion	3orel1	$$⊢\neg \phi \to \left(\left(\phi \vee \psi \vee \chi \right)\to \left(\psi \vee \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		3orass	$$⊢\left(\phi \vee \psi \vee \chi \right)\leftrightarrow \left(\phi \vee \left(\psi \vee \chi \right)\right)$$
2		orel1	$$⊢\neg \phi \to \left(\left(\phi \vee \left(\psi \vee \chi \right)\right)\to \left(\psi \vee \chi \right)\right)$$
3	1 2	biimtrid	$$⊢\neg \phi \to \left(\left(\phi \vee \psi \vee \chi \right)\to \left(\psi \vee \chi \right)\right)$$