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 ) )