Consider the following premises. For all x_9, if x_9 were a bartender, then x_9 is excited. For all x_9, if x_9 were an artist, then x_9 is playing a game. For all x_9, x_9 is an artist if and only if x_9 is excited. For all x_9, if x_9 plays a game, then x_9 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_9 for which if is not true that x_9 is an artist if and only if x_9 will play a game.
 From the fact that for all x_9, if x_9 plays a game, then x_9 is a bartender, and that for all x_9, if x_9 were a bartender, then x_9 is excited we can infer that for all x_9, if x_9 plays a game, then x_9 is excited via universal transitivity. From the fact that for all x_9, x_9 is an artist if and only if x_9 is excited we can infer that for all x_9, if x_9 were excited, then x_9 is an artist via universal biconditional elimination. From the fact that for all x_9, if x_9 plays a game, then x_9 is excited, and that for all x_9, if x_9 were excited, then x_9 is an artist we can infer that for all x_9, if x_9 plays a game, then x_9 is an artist via universal transitivity. Finally, from the fact that for all x_9, if x_9 were an artist, then x_9 is playing a game, and that for all x_9, if x_9 plays a game, then x_9 is an artist we can infer that for all x_9, x_9 is an artist if and only if x_9 will play a game via universal biconditional introduction, which contradicts that There is at least one x_9 for which if is not true that x_9 is an artist if and only if x_9 will play a game. Therefore, the answer is no.