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	$$⊢\phi \to \left(\psi \to \chi \right)$$
	sylan9.2	$$⊢\theta \to \left(\chi \to \tau \right)$$
Assertion	sylan9	$$⊢\left(\phi \wedge \theta \right)\to \left(\psi \to \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		sylan9.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		sylan9.2	$$⊢\theta \to \left(\chi \to \tau \right)$$
3	1 2	syl9	$$⊢\phi \to \left(\theta \to \left(\psi \to \tau \right)\right)$$
4	3	imp	$$⊢\left(\phi \wedge \theta \right)\to \left(\psi \to \tau \right)$$