Consider the following premises. Patricia is a soldier if and only if Patricia is playing a game. For all x_3, x_3 will play a game if and only if x_3 is a soldier. Can we infer the following from them? Patricia is playing a game if and only if Patricia is a soldier.
 From the fact that for all x_3, x_3 will play a game if and only if x_3 is a soldier we can infer that for all x_3, if x_3 plays a game, then x_3 is a soldier. From the fact that for all x_3, if x_3 plays a game, then x_3 is a soldier we can infer that if Patricia plays a game, then Patricia is a soldier. From the fact that Patricia is a soldier if and only if Patricia is playing a game we can infer that if Patricia were a soldier, then Patricia is playing a game. Finally, from the fact that if Patricia plays a game, then Patricia is a soldier, and that if Patricia were a soldier, then Patricia is playing a game we can infer that Patricia is playing a game if and only if Patricia is a soldier. Therefore, the answer is yes.