Consider the following premises. For all x_6, x_6 is not fast or x_6 is rich. For all x_6, if x_6 were rich, then x_6 is sleeping. For all x_6, if x_6 takes a plane, then x_6 is sleeping. For all x_6, if x_6 were a cashier, then x_6 is not sleeping. For all x_6, x_6 is fast or x_6 will take a plane. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 is not a cashier.
 From the fact that for all x_6, x_6 is fast or x_6 will take a plane, and that for all x_6, x_6 is not fast or x_6 is rich we can infer that for all x_6, x_6 will take a plane or x_6 is rich via universal resolution. From the fact that for all x_6, if x_6 takes a plane, then x_6 is sleeping, that for all x_6, if x_6 were rich, then x_6 is sleeping, and that for all x_6, x_6 will take a plane or x_6 is rich we can infer that for all x_6, x_6 will sleep via universal disjunction elimination. Finally, from the fact that for all x_6, if x_6 were a cashier, then x_6 is not sleeping, and that for all x_6, x_6 will sleep we can infer that for all x_6, x_6 is not a cashier via universal modus tollens. Therefore, the answer is yes.