Theorem syland

Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004)

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

Proof

Step	Hyp	Ref	Expression
1		syland.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		syland.2	$$⊢\phi \to \left(\left(\chi \wedge \theta \right)\to \tau \right)$$
3	2	expd	$$⊢\phi \to \left(\chi \to \left(\theta \to \tau \right)\right)$$
4	1 3	syld	$$⊢\phi \to \left(\psi \to \left(\theta \to \tau \right)\right)$$
5	4	impd	$$⊢\phi \to \left(\left(\psi \wedge \theta \right)\to \tau \right)$$