Consider the following premises. For all y_1, y_1 is not a cashier. For all y_1, y_1 is a cashier if and only if y_1 is excited. There is at least one y_1 for which if y_1 were a cashier, then y_1 is excited. Can we infer the following from them? For all y_1, y_1 is excited.
 From the fact that for all y_1, y_1 is a cashier if and only if y_1 is excited we can infer that for all y_1, if y_1 were excited, then y_1 is a cashier. From the fact that for all y_1, if y_1 were excited, then y_1 is a cashier, and that there is at least one y_1 for which if y_1 were a cashier, then y_1 is excited we can infer that there is at least one y_1 for which y_1 is excited if and only if y_1 is a cashier. Finally, from the fact that there is at least one y_1 for which y_1 is excited if and only if y_1 is a cashier, and that for all y_1, y_1 is not a cashier we can infer that there is at least one y_1 for which y_1 is not excited, which contradicts that For all y_1, y_1 is excited. Therefore, the answer is no.