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	⊢ ( ¬ 𝜓 → ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) → ( 𝜑 ∨ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		3orrot	⊢ ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ↔ ( 𝜓 ∨ 𝜒 ∨ 𝜑 ) )
2		3orel1	⊢ ( ¬ 𝜓 → ( ( 𝜓 ∨ 𝜒 ∨ 𝜑 ) → ( 𝜒 ∨ 𝜑 ) ) )
3		orcom	⊢ ( ( 𝜒 ∨ 𝜑 ) ↔ ( 𝜑 ∨ 𝜒 ) )
4	2 3	syl6ib	⊢ ( ¬ 𝜓 → ( ( 𝜓 ∨ 𝜒 ∨ 𝜑 ) → ( 𝜑 ∨ 𝜒 ) ) )
5	1 4	syl5bi	⊢ ( ¬ 𝜓 → ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) → ( 𝜑 ∨ 𝜒 ) ) )