Consider the following premises. There is at least one x_6 for which x_6 will not take a plane. For all x_6, x_6 is a cashier if and only if x_6 will sleep. For all x_6, if x_6 were a cashier, then x_6 is taking a plane. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which x_6 will not sleep.
 From the fact that for all x_6, if x_6 were a cashier, then x_6 is taking a plane, and that there is at least one x_6 for which x_6 will not take a plane we can infer that there is at least one x_6 for which x_6 is not a cashier via existential modus tollens. From the fact that for all x_6, x_6 is a cashier if and only if x_6 will sleep we can infer that for all x_6, if x_6 sleeps, then x_6 is a cashier via universal biconditional elimination. Finally, from the fact that for all x_6, if x_6 sleeps, then x_6 is a cashier, and that there is at least one x_6 for which x_6 is not a cashier we can infer that there is at least one x_6 for which x_6 will not sleep via existential modus tollens. Therefore, the answer is yes.