Theorem syl9

Description: A nested syllogism inference with different antecedents. (Contributed by NM, 13-May-1993) (Proof shortened by Josh Purinton, 29-Dec-2000)

	Ref	Expression
Hypotheses	syl9.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	syl9.2	⊢ ( 𝜃 → ( 𝜒 → 𝜏 ) )
Assertion	syl9	⊢ ( 𝜑 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		syl9.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		syl9.2	⊢ ( 𝜃 → ( 𝜒 → 𝜏 ) )
3	2	a1i	⊢ ( 𝜑 → ( 𝜃 → ( 𝜒 → 𝜏 ) ) )
4	1 3	syl5d	⊢ ( 𝜑 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) )