Consider the following premises. For all y, if y were a cashier, then y is cooking. For all y, y will cook if and only if y is a taxi driver. There is at least one y for which if y were a taxi driver, then y is a politician. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y for which if y were a cashier, then y is a politician.
 From the fact that for all y, y will cook if and only if y is a taxi driver we can infer that for all y, if y cooks, then y is a taxi driver via universal biconditional elimination. From the fact that for all y, if y cooks, then y is a taxi driver, and that there is at least one y for which if y were a taxi driver, then y is a politician we can infer that there is at least one y for which if y cooks, then y is a politician via existential transitivity. Finally, from the fact that for all y, if y were a cashier, then y is cooking, and that there is at least one y for which if y cooks, then y is a politician we can infer that there is at least one y for which if y were a cashier, then y is a politician via existential transitivity. Therefore, the answer is yes.