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there is no 'lavatory vent'. Its an old house, and the whole bath plumbing is just vented off the main stack (into which the pipe runs from the tub and sink drains). The stack is vertical 4" cast iron, so I don't see how that could be clogged. Isn't it more likely that there is a clog in the pipe down stream from the sink? Hence, that's why the sink tries to vent through the tub drain...
You don't have a clog or a drainage problem. You sink's venting through the tub waste because it has nowhere else to vent. Install a AAV( Air Admittance Vent) just dowstream of the lavatory trap and ree if that doesn't solfe the problem. Good luck, Tom