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Metamath Proof Explorer

Theorem jaodan

Description: Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005)

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

Proof

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