Theorem syl7

Description: A syllogism rule of inference. The first premise is used to replace the third antecedent of the second premise. (Contributed by NM, 12-Jan-1993) (Proof shortened by Wolf Lammen, 3-Aug-2012)

	Ref	Expression
Hypotheses	syl7.1	$$⊢\phi \to \psi$$
	syl7.2	$$⊢\chi \to \left(\theta \to \left(\psi \to \tau \right)\right)$$
Assertion	syl7	$$⊢\chi \to \left(\theta \to \left(\phi \to \tau \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		syl7.1	$$⊢\phi \to \psi$$
2		syl7.2	$$⊢\chi \to \left(\theta \to \left(\psi \to \tau \right)\right)$$
3	1	a1i	$$⊢\chi \to \left(\phi \to \psi \right)$$
4	3 2	syl5d	$$⊢\chi \to \left(\theta \to \left(\phi \to \tau \right)\right)$$