Consider the following premises. There is at least one x_10 for which x_10 will sleep if and only if x_10 is a bartender. For all x_10, x_10 will go running. For all x_10, x_10 will go running if and only if x_10 is a bartender. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which x_10 is a bartender.
 From the fact that there is at least one x_10 for which x_10 will sleep if and only if x_10 is a bartender we can infer that there is at least one x_10 for which if x_10 sleeps, then x_10 is a bartender via existential biconditional elimination. From the fact that for all x_10, x_10 will go running we can infer that for all x_10, x_10 will go running or x_10 will sleep via universal addition. From the fact that for all x_10, x_10 will go running if and only if x_10 is a bartender we can infer that for all x_10, if x_10 goes running, then x_10 is a bartender via universal biconditional elimination. Finally, from the fact that for all x_10, if x_10 goes running, then x_10 is a bartender, that there is at least one x_10 for which if x_10 sleeps, then x_10 is a bartender, and that for all x_10, x_10 will go running or x_10 will sleep we can infer that there is at least one x_10 for which x_10 is a bartender via existential disjunction elimination. Therefore, the answer is yes.