Theorem syldanl

Description: A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011)

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

Proof

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