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Metamath Proof Explorer

Theorem mpanlr1

Description: An inference based on modus ponens. (Contributed by NM, 30-Dec-2004) (Proof shortened by Wolf Lammen, 7-Apr-2013)

Ref Expression
Hypotheses mpanlr1.1 ⊢ 𝜓
mpanlr1.2 ⊢ ( ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) ∧ 𝜃 ) → 𝜏 )
Assertion mpanlr1 ( ( ( 𝜑 ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )

Proof

Step Hyp Ref Expression
1 mpanlr1.1 ⊢ 𝜓
2 mpanlr1.2 ⊢ ( ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) ∧ 𝜃 ) → 𝜏 )
3 1 jctl ⊢ ( 𝜒 → ( 𝜓 ∧ 𝜒 ) )
4 3 2 sylanl2 ⊢ ( ( ( 𝜑 ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )