Consider the following premises. For all z, if z sleeps, then z is a bartender. For all z, if z were a cashier, then z is sleeping. For all z, if z sleeps, then z is a cashier. For all z, if z were a bartender, then z is sleeping. For all z, z is not a cashier. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which z is a bartender.
 From the fact that for all z, if z sleeps, then z is a cashier, and that for all z, if z were a cashier, then z is sleeping we can infer that for all z, z will sleep if and only if z is a cashier via universal biconditional introduction. From the fact that for all z, if z sleeps, then z is a bartender, and that for all z, if z were a bartender, then z is sleeping we can infer that for all z, z will sleep if and only if z is a bartender via universal biconditional introduction. From the fact that for all z, z will sleep if and only if z is a cashier, and that for all z, z is not a cashier we can infer that for all z, z will not sleep via universal biconditional elimination. Finally, from the fact that for all z, z will sleep if and only if z is a bartender, and that for all z, z will not sleep we can infer that for all z, z is not a bartender via universal biconditional elimination, which contradicts that There is at least one z for which z is a bartender. Therefore, the answer is no.