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Metamath Proof Explorer

Theorem syl6mpi

Description: A syllogism inference. (Contributed by Alan Sare, 8-Jul-2011) (Proof shortened by Wolf Lammen, 13-Sep-2012)

Ref Expression
Hypotheses syl6mpi.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
syl6mpi.2 ⊢ 𝜃
syl6mpi.3 ⊢ ( 𝜒 → ( 𝜃 → 𝜏 ) )
Assertion syl6mpi ( 𝜑 → ( 𝜓 → 𝜏 ) )

Proof

Step Hyp Ref Expression
1 syl6mpi.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2 syl6mpi.2 ⊢ 𝜃
3 syl6mpi.3 ⊢ ( 𝜒 → ( 𝜃 → 𝜏 ) )
4 2 3 mpi ⊢ ( 𝜒 → 𝜏 )
5 1 4 syl6 ⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )