Consider the following premises. There is at least one x_3 for which x_3 is not rich. For all x_3, if x_3 were a bartender, then x_3 is excited. For all x_3, if x_3 were a bartender, then x_3 is rich. For all x_3, if x_3 were excited, then x_3 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_3 for which x_3 is not excited.
 From the fact that for all x_3, if x_3 were a bartender, then x_3 is rich, and that there is at least one x_3 for which x_3 is not rich we can infer that there is at least one x_3 for which x_3 is not a bartender via existential modus tollens. From the fact that for all x_3, if x_3 were excited, then x_3 is a bartender, and that for all x_3, if x_3 were a bartender, then x_3 is excited we can infer that for all x_3, x_3 is excited if and only if x_3 is a bartender via universal biconditional introduction. Finally, from the fact that for all x_3, x_3 is excited if and only if x_3 is a bartender, and that there is at least one x_3 for which x_3 is not a bartender we can infer that there is at least one x_3 for which x_3 is not excited via existential biconditional elimination. Therefore, the answer is yes.