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Metamath Proof Explorer

Theorem jaoian

Description: Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005)

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

Proof

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