I'm installing a large Kohler drop-in cast iron laundry tub in the main floor laundry toom and would like to know if this needs to be vented as I was under the impression that if you drain into a larger diameter pipe no extra vent is required.

The drain on the tub is 1-1/2" and that will go into a 2" 90-deg in the wall, run about 2 feet horizontial, then turn 90 and go down through the wall into the crawl space. The 2" line connects with the 3" waste/vent line for the upstairs master bath shower and WC.

This 3" line also carries waste from the main floor clothes washer. Without opening up the laundry room wall I don't know how the washer vents. It may have a vent that goes up and connects with the vent line for the master bath shower directly above.

If I need to add a vent, can I just go up vertically xx feet, then over to the assumed washer vent, which I'd guess would be about 4-5 feet horizontally away.

Thanks for any help. I'm under the UPC.