Consider the following premises. For all x_12, x_12 is excited if and only if x_12 will play a game. For all x_12, x_12 is not happy. There is at least one x_12 for which x_12 is happy or x_12 is excited. Can we infer the following from them? If we can, name the inference rule being used: For all x_12, x_12 will not play a game.
 From the fact that for all x_12, x_12 is excited if and only if x_12 will play a game we can infer that for all x_12, if x_12 were excited, then x_12 is playing a game via universal biconditional elimination. From the fact that there is at least one x_12 for which x_12 is happy or x_12 is excited, and that for all x_12, x_12 is not happy we can infer that there is at least one x_12 for which x_12 is excited via existential disjunctive syllogism. Finally, from the fact that for all x_12, if x_12 were excited, then x_12 is playing a game, and that there is at least one x_12 for which x_12 is excited we can infer that there is at least one x_12 for which x_12 will play a game via existential modus ponens, which contradicts that For all x_12, x_12 will not play a game. Therefore, the answer is no.