Theorem sylan9r

Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993)

	Ref	Expression
Hypotheses	sylan9r.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	sylan9r.2	$$⊢\theta \to \left(\chi \to \tau \right)$$
Assertion	sylan9r	$$⊢\left(\theta \wedge \phi \right)\to \left(\psi \to \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		sylan9r.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		sylan9r.2	$$⊢\theta \to \left(\chi \to \tau \right)$$
3	1 2	syl9r	$$⊢\theta \to \left(\phi \to \left(\psi \to \tau \right)\right)$$
4	3	imp	$$⊢\left(\theta \wedge \phi \right)\to \left(\psi \to \tau \right)$$