Theorem syl5

Description: A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. (Contributed by NM, 27-Dec-1992) (Proof shortened by Wolf Lammen, 25-May-2013)

	Ref	Expression
Hypotheses	syl5.1	$$⊢\phi \to \psi$$
	syl5.2	$$⊢\chi \to \left(\psi \to \theta \right)$$
Assertion	syl5	$$⊢\chi \to \left(\phi \to \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		syl5.1	$$⊢\phi \to \psi$$
2		syl5.2	$$⊢\chi \to \left(\psi \to \theta \right)$$
3	1 2	syl5com	$$⊢\phi \to \left(\chi \to \theta \right)$$
4	3	com12	$$⊢\chi \to \left(\phi \to \theta \right)$$