Theorem 3syld

Description: Triple syllogism deduction. Deduction associated with 3syld . (Contributed by Jeff Hankins, 4-Aug-2009)

	Ref	Expression
Hypotheses	3syld.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	3syld.2	$$⊢\phi \to \left(\chi \to \theta \right)$$
	3syld.3	$$⊢\phi \to \left(\theta \to \tau \right)$$
Assertion	3syld	$$⊢\phi \to \left(\psi \to \tau \right)$$

Proof

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