Consider the following premises. There is at least one x_15 for which if x_15 cooks, then x_15 is a mechanic. For all x_15, x_15 is a mechanic or x_15 is not a politician. For all x_15, x_15 will cook if and only if x_15 is a mechanic. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_15 for which if x_15 were a politician, then x_15 is cooking.
 From the fact that for all x_15, x_15 will cook if and only if x_15 is a mechanic we can infer that for all x_15, if x_15 were a mechanic, then x_15 is cooking via universal biconditional elimination. From the fact that for all x_15, if x_15 were a mechanic, then x_15 is cooking, and that there is at least one x_15 for which if x_15 cooks, then x_15 is a mechanic we can infer that there is at least one x_15 for which x_15 is a mechanic if and only if x_15 will cook via existential biconditional introduction. From the fact that there is at least one x_15 for which x_15 is a mechanic if and only if x_15 will cook we can infer that there is at least one x_15 for which if x_15 were a mechanic, then x_15 is cooking via existential biconditional elimination. From the fact that for all x_15, x_15 is a mechanic or x_15 is not a politician we can infer that for all x_15, if x_15 were a politician, then x_15 is a mechanic via universal rewriting. Finally, from the fact that for all x_15, if x_15 were a politician, then x_15 is a mechanic, and that there is at least one x_15 for which if x_15 were a mechanic, then x_15 is cooking we can infer that there is at least one x_15 for which if x_15 were a politician, then x_15 is cooking via existential transitivity. Therefore, the answer is yes.