Consider the following premises. For all z, if z were bored, then z is a cashier. For all z, if z were a cashier, then z is bored. For all z, z will not work. There is at least one z for which z will work if and only if z is a cashier. For all z, if z were bored, then z is a cashier. Can we infer the following from them? There is at least one z for which z is not bored.
 From the fact that for all z, if z were bored, then z is a cashier, and that for all z, if z were a cashier, then z is bored we can infer that for all z, z is bored if and only if z is a cashier. From the fact that there is at least one z for which z will work if and only if z is a cashier, and that for all z, z will not work we can infer that there is at least one z for which z is not a cashier. From the fact that for all z, z is bored if and only if z is a cashier we can infer that for all z, if z were a cashier, then z is bored. From the fact that for all z, if z were a cashier, then z is bored, and that for all z, if z were bored, then z is a cashier we can infer that for all z, z is a cashier if and only if z is bored. Finally, from the fact that for all z, z is a cashier if and only if z is bored, and that there is at least one z for which z is not a cashier we can infer that there is at least one z for which z is not bored. Therefore, the answer is yes.