First of all does your local plumbing code authority recognize the use of AAV?IPC allows AAVees,while UPC does not,anyway whether the sink+toilet are side by side or facing each other it does not matter.Establish where your cloest bend will be,it can be 3",run it horizontialy and the install a 3x1.5" wye before you connect the 3" to the 4" stack use a 4x3 no-hub or mission coupling to make that connection.Now out of the 1.5" outlet of the wye run it horizontially and the up thru the floor and the stud wall that the sink will be located,if your allowed to use an AAV you would turn a 1.5"90 into your vanity or cabinet lay a 1.5"tee on its back,the riser would be where the AAV is installed,the horiz.would be where the p-trap of the sink would be,Now if you can not use an AAV you would run the 1.5" pipe for the lav install a 1.5" san.tee about 18" off your finish floor run hoizontial length of pipe to be the waste arm for the trap of the sink,keep it less than 5' vent to trap distance.The vertical out-let of the 1.5" san. tee becomes your vent pipe,run it vertically above the flood level of the sink then you can use 90's+45's for change of direction and tie into your existing venting system or run it thru the roof but you must increase to 3" 12"above + below the roof line.What I have just described is wet venting the half-bath.





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