Consider the following premises. For all x_10, if x_10 listens to a song, then x_10 is driving a car. For all x_10, if x_10 were a soldier, then x_10 is driving a car. For all x_10, if x_10 were a soldier, then x_10 is playing a game. For all x_10, if x_10 drives a car, then x_10 is a soldier. For all x_10, if x_10 plays a game, then x_10 is listening to a song. 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 if is not true that x_10 will play a game if and only if x_10 will drive a car.
 From the fact that for all x_10, if x_10 plays a game, then x_10 is listening to a song, and that for all x_10, if x_10 listens to a song, then x_10 is driving a car we can infer that for all x_10, if x_10 plays a game, then x_10 is driving a car via universal transitivity. From the fact that for all x_10, if x_10 were a soldier, then x_10 is driving a car, and that for all x_10, if x_10 drives a car, then x_10 is a soldier we can infer that for all x_10, x_10 is a soldier if and only if x_10 will drive a car via universal biconditional introduction. From the fact that for all x_10, x_10 is a soldier if and only if x_10 will drive a car we can infer that for all x_10, if x_10 drives a car, then x_10 is a soldier via universal biconditional elimination. From the fact that for all x_10, if x_10 drives a car, then x_10 is a soldier, and that for all x_10, if x_10 were a soldier, then x_10 is playing a game we can infer that for all x_10, if x_10 drives a car, then x_10 is playing a game via universal transitivity. Finally, from the fact that for all x_10, if x_10 plays a game, then x_10 is driving a car, and that for all x_10, if x_10 drives a car, then x_10 is playing a game we can infer that for all x_10, x_10 will play a game if and only if x_10 will drive a car via universal biconditional introduction, which contradicts that There is at least one x_10 for which if is not true that x_10 will play a game if and only if x_10 will drive a car. Therefore, the answer is no.