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Metamath Proof Explorer

Theorem jaoa

Description: Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008)

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

Proof

Step Hyp Ref Expression
1 jaao.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2 jaao.2 ⊢ ( 𝜃 → ( 𝜏 → 𝜒 ) )
3 1 adantrd ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜏 ) → 𝜒 ) )
4 2 adantld ⊢ ( 𝜃 → ( ( 𝜓 ∧ 𝜏 ) → 𝜒 ) )
5 3 4 jaoi ⊢ ( ( 𝜑 ∨ 𝜃 ) → ( ( 𝜓 ∧ 𝜏 ) → 𝜒 ) )