Consider the following premises. For all z, if z were a police officer, then z is a cashier. There is at least one z for which z is a lawyer if and only if z is a police officer. For all z, if z were a cashier, then z is a police officer. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which if z were a cashier, then z is a lawyer.
 From the fact that for all z, if z were a police officer, then z is a cashier, and that for all z, if z were a cashier, then z is a police officer we can infer that for all z, z is a police officer if and only if z is a cashier via universal biconditional introduction. From the fact that there is at least one z for which z is a lawyer if and only if z is a police officer we can infer that there is at least one z for which if z were a police officer, then z is a lawyer via existential biconditional elimination. From the fact that for all z, z is a police officer if and only if z is a cashier we can infer that for all z, if z were a cashier, then z is a police officer via universal biconditional elimination. Finally, from the fact that for all z, if z were a cashier, then z is a police officer, and that there is at least one z for which if z were a police officer, then z is a lawyer we can infer that there is at least one z for which if z were a cashier, then z is a lawyer via existential transitivity. Therefore, the answer is yes.