Theorem sylancom

Description: Syllogism inference with commutation of antecedents. (Contributed by NM, 2-Jul-2008)

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

Proof

Step	Hyp	Ref	Expression
1		sylancom.1	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
2		sylancom.2	$$⊢\left(\chi \wedge \psi \right)\to \theta$$
3		simpr	$$⊢\left(\phi \wedge \psi \right)\to \psi$$
4	1 3 2	syl2anc	$$⊢\left(\phi \wedge \psi \right)\to \theta$$