Consider the following premises. For all x_4, if x_4 were a writer, then x_4 is playing a game. There is at least one x_4 for which if x_4 were a cashier, then x_4 is a writer. For all x_4, if x_4 plays a game, then x_4 is a cashier. Can we infer the following from them? There is at least one x_4 for which if x_4 were a writer, then x_4 is playing a game.
 From the fact that for all x_4, if x_4 plays a game, then x_4 is a cashier, and that there is at least one x_4 for which if x_4 were a cashier, then x_4 is a writer we can infer that there is at least one x_4 for which if x_4 plays a game, then x_4 is a writer. From the fact that for all x_4, if x_4 were a writer, then x_4 is playing a game, and that there is at least one x_4 for which if x_4 plays a game, then x_4 is a writer we can infer that there is at least one x_4 for which x_4 is a writer if and only if x_4 will play a game. Finally, from the fact that there is at least one x_4 for which x_4 is a writer if and only if x_4 will play a game we can infer that there is at least one x_4 for which if x_4 were a writer, then x_4 is playing a game. Therefore, the answer is yes.