Consider the following premises. There is at least one x_5 for which x_5 is sad. For all x_5, if x_5 plays a game, then x_5 is playing tennis. For all x_5, if x_5 plays tennis, then x_5 is playing a game. For all x_5, if x_5 plays a game, then x_5 is not sad. 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 not play tennis.
 From the fact that for all x_5, if x_5 plays a game, then x_5 is not sad, and that there is at least one x_5 for which x_5 is sad we can infer that there is at least one x_5 for which x_5 will not play a game via existential modus tollens. From the fact that for all x_5, if x_5 plays tennis, then x_5 is playing a game, and that for all x_5, if x_5 plays a game, then x_5 is playing tennis we can infer that for all x_5, x_5 will play tennis if and only if x_5 will play a game via universal biconditional introduction. Finally, from the fact that for all x_5, x_5 will play tennis if and only if x_5 will play a game, and that there is at least one x_5 for which x_5 will not play a game we can infer that there is at least one x_5 for which x_5 will not play tennis via existential biconditional elimination. Therefore, the answer is yes.