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Metamath Proof Explorer

Theorem jaao

Description: Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999)

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

Proof

Step Hyp Ref Expression
1 jaao.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2 jaao.2 ⊢ ( 𝜃 → ( 𝜏 → 𝜒 ) )
3 1 adantr ⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 → 𝜒 ) )
4 2 adantl ⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜏 → 𝜒 ) )
5 3 4 jaod ⊢ ( ( 𝜑 ∧ 𝜃 ) → ( ( 𝜓 ∨ 𝜏 ) → 𝜒 ) )