I'm hanging a lightweight bathroom storage cabinet (I'd estimate 30-40 lbs max when full). The back has three horizontal wood rails, and there happens to be one stud that runs almost perfectly centered across the cabinet width.

My plan is to drive one 2-1/2" cabinet screw through each rail into that stud which would be three screws total, vertically distributed across the height of the cabinet without any wall anchors on the sides.

Does this seem solid, or would you add anchors on the sides as well?