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) (Proof shortened by Eric Schmidt, 8-Oct-2025)

	Ref	Expression
Assertion	3orel2	⊢ ( ¬ 𝜓 → ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) → ( 𝜑 ∨ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		3orcoma	⊢ ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ↔ ( 𝜓 ∨ 𝜑 ∨ 𝜒 ) )
2		3orel1	⊢ ( ¬ 𝜓 → ( ( 𝜓 ∨ 𝜑 ∨ 𝜒 ) → ( 𝜑 ∨ 𝜒 ) ) )
3	1 2	biimtrid	⊢ ( ¬ 𝜓 → ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) → ( 𝜑 ∨ 𝜒 ) ) )