Consider the following premises. There is at least one y for which if y plays a game, then y is a soldier. For all y, y will play a game if and only if y is a soldier. For all y, if y were a soldier, then y is playing a game. Can we infer the following from them? There is at least one y for which if y plays a game, then y is a soldier.
That is one of the premises. Therefore, the answer is yes.