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Metamath Proof Explorer

Theorem mpan9

Description: Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008) (Proof shortened by Andrew Salmon, 7-May-2011)

Ref Expression
Hypotheses mpan9.1 ⊢ ( 𝜑 → 𝜓 )
mpan9.2 ⊢ ( 𝜒 → ( 𝜓 → 𝜃 ) )
Assertion mpan9 ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )

Proof

Step Hyp Ref Expression
1 mpan9.1 ⊢ ( 𝜑 → 𝜓 )
2 mpan9.2 ⊢ ( 𝜒 → ( 𝜓 → 𝜃 ) )
3 1 2 syl5 ⊢ ( 𝜒 → ( 𝜑 → 𝜃 ) )
4 3 impcom ⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )