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Metamath Proof Explorer

Theorem syl10

Description: A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011)

Ref Expression
Hypotheses syl10.1 ⊢ φ → ψ → χ
syl10.2 ⊢ φ → ψ → θ → τ
syl10.3 ⊢ χ → τ → η
Assertion syl10 ⊢ φ → ψ → θ → η

Proof

Step Hyp Ref Expression
1 syl10.1 ⊢ φ → ψ → χ
2 syl10.2 ⊢ φ → ψ → θ → τ
3 syl10.3 ⊢ χ → τ → η
4 1 3 syl6 ⊢ φ → ψ → τ → η
5 2 4 syldd ⊢ φ → ψ → θ → η