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Metamath Proof Explorer

Theorem sylsyld

Description: A double syllogism inference. (Contributed by Alan Sare, 20-Apr-2011)

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

Proof

Step Hyp Ref Expression
1 sylsyld.1 ⊢ ( 𝜑 → 𝜓 )
2 sylsyld.2 ⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
3 sylsyld.3 ⊢ ( 𝜓 → ( 𝜃 → 𝜏 ) )
4 1 3 syl ⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
5 2 4 syld ⊢ ( 𝜑 → ( 𝜒 → 𝜏 ) )