Consider the following premises. For all x_10, x_10 will play a game if and only if x_10 is a police officer. There is at least one x_10 for which x_10 is not excited. For all x_10, x_10 is excited if and only if x_10 will play a game. Can we infer the following from them? If we can, name the inference rule being used: For all x_10, x_10 is a police officer.
 From the fact that for all x_10, x_10 is excited if and only if x_10 will play a game, and that there is at least one x_10 for which x_10 is not excited we can infer that there is at least one x_10 for which x_10 will not play a game via existential biconditional elimination. From the fact that for all x_10, x_10 will play a game if and only if x_10 is a police officer we can infer that for all x_10, if x_10 were a police officer, then x_10 is playing a game via universal biconditional elimination. Finally, from the fact that for all x_10, if x_10 were a police officer, then x_10 is playing a game, and that there is at least one x_10 for which x_10 will not play a game we can infer that there is at least one x_10 for which x_10 is not a police officer via existential modus tollens, which contradicts that For all x_10, x_10 is a police officer. Therefore, the answer is no.