Theorem sylan9

Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993) (Proof shortened by Andrew Salmon, 7-May-2011)

	Ref	Expression
Hypotheses	sylan9.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	sylan9.2	⊢ ( 𝜃 → ( 𝜒 → 𝜏 ) )
Assertion	sylan9	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 → 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		sylan9.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		sylan9.2	⊢ ( 𝜃 → ( 𝜒 → 𝜏 ) )
3	1 2	syl9	⊢ ( 𝜑 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) )
4	3	imp	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 → 𝜏 ) )