Consider the following premises. For all x_5, if x_5 cooks, then x_5 is a cashier. For all x_5, if x_5 were a cashier, then x_5 is cooking. There is at least one x_5 for which if x_5 cooks, then x_5 is a cashier. Can we infer the following from them? There is at least one x_5 for which x_5 is a cashier if and only if x_5 will cook.
 From the fact that for all x_5, if x_5 were a cashier, then x_5 is cooking, and that for all x_5, if x_5 cooks, then x_5 is a cashier we can infer that for all x_5, x_5 is a cashier if and only if x_5 will cook. From the fact that for all x_5, x_5 is a cashier if and only if x_5 will cook we can infer that for all x_5, if x_5 were a cashier, then x_5 is cooking. Finally, from the fact that for all x_5, if x_5 were a cashier, then x_5 is cooking, and that there is at least one x_5 for which if x_5 cooks, then x_5 is a cashier we can infer that there is at least one x_5 for which x_5 is a cashier if and only if x_5 will cook. Therefore, the answer is yes.