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Metamath Proof Explorer

Theorem 3jaoian

Description: Disjunction of three antecedents (inference). (Contributed by NM, 14-Oct-2005)

Ref Expression
Hypotheses 3jaoian.1 ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
3jaoian.2 ⊢ ( ( 𝜃 ∧ 𝜓 ) → 𝜒 )
3jaoian.3 ⊢ ( ( 𝜏 ∧ 𝜓 ) → 𝜒 )
Assertion 3jaoian ( ( ( 𝜑 ∨ 𝜃 ∨ 𝜏 ) ∧ 𝜓 ) → 𝜒 )

Proof

Step Hyp Ref Expression
1 3jaoian.1 ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2 3jaoian.2 ⊢ ( ( 𝜃 ∧ 𝜓 ) → 𝜒 )
3 3jaoian.3 ⊢ ( ( 𝜏 ∧ 𝜓 ) → 𝜒 )
4 1 ex ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
5 2 ex ⊢ ( 𝜃 → ( 𝜓 → 𝜒 ) )
6 3 ex ⊢ ( 𝜏 → ( 𝜓 → 𝜒 ) )
7 4 5 6 3jaoi ⊢ ( ( 𝜑 ∨ 𝜃 ∨ 𝜏 ) → ( 𝜓 → 𝜒 ) )
8 7 imp ⊢ ( ( ( 𝜑 ∨ 𝜃 ∨ 𝜏 ) ∧ 𝜓 ) → 𝜒 )