Theorem syli

Description: Syllogism inference with common nested antecedent. (Contributed by NM, 4-Nov-2004)

	Ref	Expression
Hypotheses	syli.1	⊢ ( 𝜓 → ( 𝜑 → 𝜒 ) )
	syli.2	⊢ ( 𝜒 → ( 𝜑 → 𝜃 ) )
Assertion	syli	⊢ ( 𝜓 → ( 𝜑 → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		syli.1	⊢ ( 𝜓 → ( 𝜑 → 𝜒 ) )
2		syli.2	⊢ ( 𝜒 → ( 𝜑 → 𝜃 ) )
3	2	com12	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
4	1 3	sylcom	⊢ ( 𝜓 → ( 𝜑 → 𝜃 ) )