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Metamath Proof Explorer

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 )