Consider the following premises. For all z, if z were a politician, then z is taking a plane. For all z, if z takes a plane, then z is a politician. There is at least one z for which if z were a politician, then z is a cashier. For all z, if z were a cashier, then z is a politician. For all z, z will not take a plane. 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 not a cashier.
 From the fact that there is at least one z for which if z were a politician, then z is a cashier, and that for all z, if z were a cashier, then z is a politician we can infer that there is at least one z for which z is a politician if and only if z is a cashier via existential biconditional introduction. From the fact that for all z, if z were a politician, then z is taking a plane, and that for all z, if z takes a plane, then z is a politician we can infer that for all z, z is a politician if and only if z will take a plane via universal biconditional introduction. From the fact that for all z, z is a politician if and only if z will take a plane, and that for all z, z will not take a plane we can infer that for all z, z is not a politician via universal biconditional elimination. Finally, from the fact that there is at least one z for which z is a politician if and only if z is a cashier, and that for all z, z is not a politician we can infer that there is at least one z for which z is not a cashier via existential biconditional elimination. Therefore, the answer is yes.