Consider the following premises. For all x_8, x_8 is a lawyer if and only if x_8 will sleep. There is at least one x_8 for which x_8 will cook or x_8 is a lawyer. For all x_8, x_8 will not cook. 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 will sleep.
 From the fact that there is at least one x_8 for which x_8 will cook or x_8 is a lawyer, and that for all x_8, x_8 will not cook we can infer that there is at least one x_8 for which x_8 is a lawyer via existential disjunctive syllogism. From the fact that for all x_8, x_8 is a lawyer if and only if x_8 will sleep we can infer that for all x_8, if x_8 were a lawyer, then x_8 is sleeping via universal biconditional elimination. Finally, from the fact that for all x_8, if x_8 were a lawyer, then x_8 is sleeping, and that there is at least one x_8 for which x_8 is a lawyer we can infer that there is at least one x_8 for which x_8 will sleep via existential modus ponens. Therefore, the answer is yes.