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Metamath Proof Explorer

Theorem mpan2i

Description: An inference based on modus ponens. (Contributed by NM, 10-Apr-1994) (Proof shortened by Wolf Lammen, 19-Nov-2012)

Ref Expression
Hypotheses mpan2i.1 ⊢ 𝜒
mpan2i.2 ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) )
Assertion mpan2i ( 𝜑 → ( 𝜓 → 𝜃 ) )

Proof

Step Hyp Ref Expression
1 mpan2i.1 ⊢ 𝜒
2 mpan2i.2 ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) )
3 1 a1i ⊢ ( 𝜑 → 𝜒 )
4 3 2 mpan2d ⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )