Consider the following premises. For all x_10, if x_10 were a janitor, then x_10 is a bartender. For all x_10, if x_10 plays a game, then x_10 is a janitor. For all x_10, x_10 is a bartender or x_10 is not a politician. For all x_10, x_10 is a politician or x_10 will play a game. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which neither x_10 is a bartender nor x_10 will climb a mountain.
 From the fact that for all x_10, x_10 is a bartender or x_10 is not a politician we can infer that for all x_10, if x_10 were a politician, then x_10 is a bartender via universal rewriting. From the fact that for all x_10, if x_10 plays a game, then x_10 is a janitor, and that for all x_10, if x_10 were a janitor, then x_10 is a bartender we can infer that for all x_10, if x_10 plays a game, then x_10 is a bartender via universal transitivity. From the fact that for all x_10, if x_10 were a politician, then x_10 is a bartender, that for all x_10, if x_10 plays a game, then x_10 is a bartender, and that for all x_10, x_10 is a politician or x_10 will play a game we can infer that for all x_10, x_10 is a bartender via universal disjunction elimination. Finally, from the fact that for all x_10, x_10 is a bartender we can infer that for all x_10, x_10 is a bartender or x_10 will climb a mountain via universal addition, which contradicts that There is at least one x_10 for which neither x_10 is a bartender nor x_10 will climb a mountain. Therefore, the answer is no.