Consider the following premises. For all y, if y were a mechanic, then y is running. For all y, if y drinks water, then y is a mechanic. There is at least one y for which y will drink water or y will not go running. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y for which y will go running if and only if y will drink water.
 From the fact that there is at least one y for which y will drink water or y will not go running we can infer that there is at least one y for which if y goes running, then y is drinking water via existential rewriting. From the fact that for all y, if y drinks water, then y is a mechanic, and that for all y, if y were a mechanic, then y is running we can infer that for all y, if y drinks water, then y is running via universal transitivity. Finally, from the fact that there is at least one y for which if y goes running, then y is drinking water, and that for all y, if y drinks water, then y is running we can infer that there is at least one y for which y will go running if and only if y will drink water via existential biconditional introduction. Therefore, the answer is yes.