Consider the following premises. For all x_6, if x_6 were a bartender, then x_6 is making tea. For all x_6, x_6 will not play a game. For all x_6, x_6 will play a game if and only if x_6 is a bartender. For all x_6, if x_6 makes tea, then x_6 is playing a game. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 is not a bartender.
 From the fact that for all x_6, if x_6 were a bartender, then x_6 is making tea, and that for all x_6, if x_6 makes tea, then x_6 is playing a game we can infer that for all x_6, if x_6 were a bartender, then x_6 is playing a game via universal transitivity. From the fact that for all x_6, x_6 will play a game if and only if x_6 is a bartender we can infer that for all x_6, if x_6 plays a game, then x_6 is a bartender via universal biconditional elimination. From the fact that for all x_6, if x_6 plays a game, then x_6 is a bartender, and that for all x_6, if x_6 were a bartender, then x_6 is playing a game we can infer that for all x_6, x_6 will play a game if and only if x_6 is a bartender via universal biconditional introduction. Finally, from the fact that for all x_6, x_6 will play a game if and only if x_6 is a bartender, and that for all x_6, x_6 will not play a game we can infer that for all x_6, x_6 is not a bartender via universal biconditional elimination. Therefore, the answer is yes.