Hi I need advice on some bathroom plumbing. I have the national plumbing handbook in front of me and this is what i've come up with. I'd like to get some insight on if i planned everything out correctly. Keep in mind the main 3" house drain is sitting under the floor about 6-8inches. It already has a kitchen and clothes washer further up stream and another bathroom downstream.

{{!gwi}}Do you think i need to extend a vent from behind my future sink and tie it into the 2" stack? or is it close enough to not deem it necessary.