Consider the following premises. For all x_8, x_8 will go running. For all x_8, x_8 will sleep if and only if x_8 will go running. There is at least one x_8 for which if x_8 were rich, then x_8 is not sleeping. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_8 for which x_8 is not rich.
 From the fact that for all x_8, x_8 will sleep if and only if x_8 will go running we can infer that for all x_8, if x_8 goes running, then x_8 is sleeping via universal biconditional elimination. From the fact that for all x_8, if x_8 goes running, then x_8 is sleeping, and that for all x_8, x_8 will go running we can infer that for all x_8, x_8 will sleep via universal modus ponens. Finally, from the fact that there is at least one x_8 for which if x_8 were rich, then x_8 is not sleeping, and that for all x_8, x_8 will sleep we can infer that there is at least one x_8 for which x_8 is not rich via existential modus tollens. Therefore, the answer is yes.