Consider the following premises. For all x_4, if x_4 were a bartender, then x_4 is playing a game. For all x_4, x_4 will play a game if and only if x_4 is a mechanic. For all x_4, x_4 is not a mechanic. Can we infer the following from them? If we can, name the inference rule being used: For all x_4, x_4 is not a bartender.
 From the fact that for all x_4, x_4 will play a game if and only if x_4 is a mechanic, and that for all x_4, x_4 is not a mechanic we can infer that for all x_4, x_4 will not play a game via universal biconditional elimination. Finally, from the fact that for all x_4, if x_4 were a bartender, then x_4 is playing a game, and that for all x_4, x_4 will not play a game we can infer that for all x_4, x_4 is not a bartender via universal modus tollens. Therefore, the answer is yes.