Theorem syl10

Description: A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011)

	Ref	Expression
Hypotheses	syl10.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	syl10.2	$$⊢\phi \to \left(\psi \to \left(\theta \to \tau \right)\right)$$
	syl10.3	$$⊢\chi \to \left(\tau \to \eta \right)$$
Assertion	syl10	$$⊢\phi \to \left(\psi \to \left(\theta \to \eta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		syl10.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		syl10.2	$$⊢\phi \to \left(\psi \to \left(\theta \to \tau \right)\right)$$
3		syl10.3	$$⊢\chi \to \left(\tau \to \eta \right)$$
4	1 3	syl6	$$⊢\phi \to \left(\psi \to \left(\tau \to \eta \right)\right)$$
5	2 4	syldd	$$⊢\phi \to \left(\psi \to \left(\theta \to \eta \right)\right)$$