Consider the following premises. For all x_7, if x_7 were excited, then x_7 is cooking. For all x_7, x_7 is a politician or x_7 is not a cashier. There is at least one x_7 for which if x_7 were a politician, then x_7 is excited. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_7 for which if x_7 were a cashier, then x_7 is cooking.
 From the fact that for all x_7, x_7 is a politician or x_7 is not a cashier we can infer that for all x_7, if x_7 were a cashier, then x_7 is a politician via universal rewriting. From the fact that there is at least one x_7 for which if x_7 were a politician, then x_7 is excited, and that for all x_7, if x_7 were excited, then x_7 is cooking we can infer that there is at least one x_7 for which if x_7 were a politician, then x_7 is cooking via existential transitivity. Finally, from the fact that for all x_7, if x_7 were a cashier, then x_7 is a politician, and that there is at least one x_7 for which if x_7 were a politician, then x_7 is cooking we can infer that there is at least one x_7 for which if x_7 were a cashier, then x_7 is cooking via existential transitivity. Therefore, the answer is yes.