My new sink is too deep for the drain. How can I plumb it to lower the p pipe?
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What I'd do to fix this, considering you can't chop the tailpiece down without removing the dishwasher inlet, would be to insert a DWV-PVC pipe nipple of some sort between the outlet bend and the P-trap itself. This should be fine as long as you don't exceed the 24" distance limit prescribed in IPC 1002.1 (inapplicable exceptions omitted, erratum applied):
Each plumbing fixture shall be separately trapped by a liquid-seal trap, except as otherwise permitted by this code. The vertical distance from the fixture outlet to the trap weir shall not exceed 24 inches (610 mm), and the horizontal distance shall not exceed 30 inches (762mm) measured from the centerline of the fixture outlet to the centerline of the inlet of the trap. The height of a clothes washer standpipe above a trap shall conform to Section 802.3.3. A fixture shall not be double trapped.