Theorem 3ori

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

	Ref	Expression
Hypothesis	3ori.1	⊢ ( 𝜑 ∨ 𝜓 ∨ 𝜒 )
Assertion	3ori	⊢ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		3ori.1	⊢ ( 𝜑 ∨ 𝜓 ∨ 𝜒 )
2		ioran	⊢ ( ¬ ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ 𝜑 ∧ ¬ 𝜓 ) )
3		df-3or	⊢ ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) )
4	1 3	mpbi	⊢ ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 )
5	4	ori	⊢ ( ¬ ( 𝜑 ∨ 𝜓 ) → 𝜒 )
6	2 5	sylbir	⊢ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) → 𝜒 )