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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