Theorem syl6an

Description: A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011)

	Ref	Expression
Hypotheses	syl6an.1	$$⊢\phi \to \psi$$
	syl6an.2	$$⊢\phi \to \left(\chi \to \theta \right)$$
	syl6an.3	$$⊢\left(\psi \wedge \theta \right)\to \tau$$
Assertion	syl6an	$$⊢\phi \to \left(\chi \to \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		syl6an.1	$$⊢\phi \to \psi$$
2		syl6an.2	$$⊢\phi \to \left(\chi \to \theta \right)$$
3		syl6an.3	$$⊢\left(\psi \wedge \theta \right)\to \tau$$
4	3	ex	$$⊢\psi \to \left(\theta \to \tau \right)$$
5	1 2 4	sylsyld	$$⊢\phi \to \left(\chi \to \tau \right)$$