Consider the following premises. For all x_3, if x_3 were a bartender, then x_3 is playing a game. For all x_3, if x_3 were a bartender, then x_3 is running. For all x_3, if x_3 plays a game, then x_3 is a bartender. There is at least one x_3 for which x_3 will not go running. 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 will not play a game.
 From the fact that for all x_3, if x_3 were a bartender, then x_3 is running, and that there is at least one x_3 for which x_3 will not go running 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 plays a game, then x_3 is a bartender, and that for all x_3, if x_3 were a bartender, then x_3 is playing a game we can infer that for all x_3, x_3 will play a game if and only if x_3 is a bartender via universal biconditional introduction. Finally, from the fact that for all x_3, x_3 will play a game 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 will not play a game via existential biconditional elimination. Therefore, the answer is yes.