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Metamath Proof Explorer

Theorem mpanl2

Description: An inference based on modus ponens. (Contributed by NM, 16-Aug-1994) (Proof shortened by Andrew Salmon, 7-May-2011)

Ref Expression
Hypotheses mpanl2.1 ⊢ 𝜓
mpanl2.2 ⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 )
Assertion mpanl2 ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )

Proof

Step Hyp Ref Expression
1 mpanl2.1 ⊢ 𝜓
2 mpanl2.2 ⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 )
3 1 jctr ⊢ ( 𝜑 → ( 𝜑 ∧ 𝜓 ) )
4 3 2 sylan ⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )