Theorem 3orel2

Description: Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Assertion	3orel2	$$⊢\neg \psi \to \left(\left(\phi \vee \psi \vee \chi \right)\to \left(\phi \vee \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		3orrot	$$⊢\left(\phi \vee \psi \vee \chi \right)\leftrightarrow \left(\psi \vee \chi \vee \phi \right)$$
2		3orel1	$$⊢\neg \psi \to \left(\left(\psi \vee \chi \vee \phi \right)\to \left(\chi \vee \phi \right)\right)$$
3		orcom	$$⊢\left(\chi \vee \phi \right)\leftrightarrow \left(\phi \vee \chi \right)$$
4	2 3	syl6ib	$$⊢\neg \psi \to \left(\left(\psi \vee \chi \vee \phi \right)\to \left(\phi \vee \chi \right)\right)$$
5	1 4	syl5bi	$$⊢\neg \psi \to \left(\left(\phi \vee \psi \vee \chi \right)\to \left(\phi \vee \chi \right)\right)$$