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Metamath Proof Explorer

Theorem 3jaoi

Description: Disjunction of three antecedents (inference). (Contributed by NM, 12-Sep-1995)

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

Proof

Step Hyp Ref Expression
1 3jaoi.1 ⊢ ( 𝜑 → 𝜓 )
2 3jaoi.2 ⊢ ( 𝜒 → 𝜓 )
3 3jaoi.3 ⊢ ( 𝜃 → 𝜓 )
4 1 2 3 3pm3.2i ⊢ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜓 ) ∧ ( 𝜃 → 𝜓 ) )
5 3jao ⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜓 ) ∧ ( 𝜃 → 𝜓 ) ) → ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 ) )
6 4 5 ax-mp ⊢ ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 )