Consider the following premises. For all x_10, if x_10 were a police officer, then x_10 is playing a game. For all x_10, if x_10 plays a game, then x_10 is playing tennis. For all x_10, if x_10 plays tennis, then x_10 is a soldier. There is at least one x_10 for which x_10 will not play tennis. For all x_10, if x_10 were a soldier, then x_10 is a police officer. 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 x_10 is not a soldier.
 From the fact that for all x_10, if x_10 were a police officer, then x_10 is playing a game, and that for all x_10, if x_10 plays a game, then x_10 is playing tennis we can infer that for all x_10, if x_10 were a police officer, then x_10 is playing tennis via universal transitivity. From the fact that for all x_10, if x_10 were a soldier, then x_10 is a police officer, and that for all x_10, if x_10 were a police officer, then x_10 is playing tennis we can infer that for all x_10, if x_10 were a soldier, then x_10 is playing tennis via universal transitivity. From the fact that for all x_10, if x_10 were a soldier, then x_10 is playing tennis, and that for all x_10, if x_10 plays tennis, then x_10 is a soldier we can infer that for all x_10, x_10 is a soldier if and only if x_10 will play tennis via universal biconditional introduction. From the fact that for all x_10, x_10 is a soldier if and only if x_10 will play tennis we can infer that for all x_10, if x_10 were a soldier, then x_10 is playing tennis via universal biconditional elimination. Finally, from the fact that for all x_10, if x_10 were a soldier, then x_10 is playing tennis, and that there is at least one x_10 for which x_10 will not play tennis we can infer that there is at least one x_10 for which x_10 is not a soldier via existential modus tollens. Therefore, the answer is yes.