Consider the following premises. For all x_5, if x_5 writes a letter, then x_5 is not playing squash. For all x_5, if x_5 plays a game, then x_5 is writing a letter. For all x_5, if x_5 writes a letter, then x_5 is playing a game. For all x_5, x_5 will play squash. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which x_5 will play a game.
 From the fact that for all x_5, if x_5 writes a letter, then x_5 is playing a game, and that for all x_5, if x_5 plays a game, then x_5 is writing a letter we can infer that for all x_5, x_5 will write a letter if and only if x_5 will play a game via universal biconditional introduction. From the fact that for all x_5, if x_5 writes a letter, then x_5 is not playing squash, and that for all x_5, x_5 will play squash we can infer that for all x_5, x_5 will not write a letter via universal modus tollens. Finally, from the fact that for all x_5, x_5 will write a letter if and only if x_5 will play a game, and that for all x_5, x_5 will not write a letter we can infer that for all x_5, x_5 will not play a game via universal biconditional elimination, which contradicts that There is at least one x_5 for which x_5 will play a game. Therefore, the answer is no.