Theorem sylcom

Description: Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004) (Proof shortened by Mel L. O'Cat, 2-Feb-2006) (Proof shortened by Stefan Allan, 23-Feb-2006)

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

Proof

Step	Hyp	Ref	Expression
1		sylcom.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		sylcom.2	$$⊢\psi \to \left(\chi \to \theta \right)$$
3	2	a2i	$$⊢\left(\psi \to \chi \right)\to \left(\psi \to \theta \right)$$
4	1 3	syl	$$⊢\phi \to \left(\psi \to \theta \right)$$