Consider the following premises. For all x, if x sleeps, then x is rich. For all x, x will sleep or x will cook. For all x, if x cooks, then x is rich. There is at least one x for which if x were rich, then x is sleeping. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which x is rich.
 From the fact that there is at least one x for which if x were rich, then x is sleeping, and that for all x, if x sleeps, then x is rich we can infer that there is at least one x for which x is rich if and only if x will sleep via existential biconditional introduction. From the fact that there is at least one x for which x is rich if and only if x will sleep we can infer that there is at least one x for which if x sleeps, then x is rich via existential biconditional elimination. Finally, from the fact that there is at least one x for which if x sleeps, then x is rich, that for all x, if x cooks, then x is rich, and that for all x, x will sleep or x will cook we can infer that there is at least one x for which x is rich via existential disjunction elimination. Therefore, the answer is yes.