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