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Metamath Proof Explorer

Theorem 3jaodan

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

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

Proof

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