Consider the following premises. For all x_10, if x_10 takes a plane, then x_10 is excited. For all x_10, if x_10 plays a game, then x_10 is excited. For all x_10, if x_10 drives a car, then x_10 is playing a game. For all x_10, x_10 will take a plane or x_10 will drive a car. There is at least one x_10 for which x_10 is excited if and only if x_10 will play tennis. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which x_10 will play tennis.
 From the fact that for all x_10, if x_10 drives a car, then x_10 is playing a game, and that for all x_10, if x_10 plays a game, then x_10 is excited we can infer that for all x_10, if x_10 drives a car, then x_10 is excited via universal transitivity. From the fact that for all x_10, if x_10 takes a plane, then x_10 is excited, that for all x_10, if x_10 drives a car, then x_10 is excited, and that for all x_10, x_10 will take a plane or x_10 will drive a car we can infer that for all x_10, x_10 is excited via universal disjunction elimination. From the fact that there is at least one x_10 for which x_10 is excited if and only if x_10 will play tennis we can infer that there is at least one x_10 for which if x_10 were excited, then x_10 is playing tennis via existential biconditional elimination. Finally, from the fact that there is at least one x_10 for which if x_10 were excited, then x_10 is playing tennis, and that for all x_10, x_10 is excited we can infer that there is at least one x_10 for which x_10 will play tennis via existential modus ponens. Therefore, the answer is yes.