Consider the following premises. For all y_2, if y_2 plays a game, then y_2 is playing squash. For all y_2, if y_2 plays squash, then y_2 is a soldier. For all y_2, if y_2 were a soldier, then y_2 is playing a game. There is at least one y_2 for which if y_2 plays a game, then y_2 is a soldier. Can we infer the following from them? If we can, name the inference rule being used: For all y_2, if is not true that y_2 is a soldier if and only if y_2 will play a game.
 From the fact that for all y_2, if y_2 were a soldier, then y_2 is playing a game, and that there is at least one y_2 for which if y_2 plays a game, then y_2 is a soldier we can infer that there is at least one y_2 for which y_2 is a soldier if and only if y_2 will play a game via existential biconditional introduction. From the fact that there is at least one y_2 for which y_2 is a soldier if and only if y_2 will play a game we can infer that there is at least one y_2 for which if y_2 were a soldier, then y_2 is playing a game via existential biconditional elimination. From the fact that for all y_2, if y_2 plays a game, then y_2 is playing squash, and that for all y_2, if y_2 plays squash, then y_2 is a soldier we can infer that for all y_2, if y_2 plays a game, then y_2 is a soldier via universal transitivity. Finally, from the fact that there is at least one y_2 for which if y_2 were a soldier, then y_2 is playing a game, and that for all y_2, if y_2 plays a game, then y_2 is a soldier we can infer that there is at least one y_2 for which y_2 is a soldier if and only if y_2 will play a game via existential biconditional introduction, which contradicts that For all y_2, if is not true that y_2 is a soldier if and only if y_2 will play a game. Therefore, the answer is no.