Theorem syldd

Description: Nested syllogism deduction. Deduction associated with syld . Double deduction associated with syl . (Contributed by NM, 12-Dec-2004) (Proof shortened by Wolf Lammen, 11-May-2013)

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

Proof

Step	Hyp	Ref	Expression
1		syldd.1	$$⊢\phi \to \left(\psi \to \left(\chi \to \theta \right)\right)$$
2		syldd.2	$$⊢\phi \to \left(\psi \to \left(\theta \to \tau \right)\right)$$
3		imim2	$$⊢\left(\theta \to \tau \right)\to \left(\left(\chi \to \theta \right)\to \left(\chi \to \tau \right)\right)$$
4	2 1 3	syl6c	$$⊢\phi \to \left(\psi \to \left(\chi \to \tau \right)\right)$$