Consider the following premises. For all x_8, if x_8 were a nurse, then x_8 is playing a game. For all x_8, x_8 will write a letter if and only if x_8 is a nurse. There is at least one x_8 for which if x_8 plays a game, then x_8 is writing a letter. Can we infer the following from them? There is at least one x_8 for which x_8 will write a letter if and only if x_8 is a nurse.
 From the fact that for all x_8, x_8 will write a letter if and only if x_8 is a nurse we can infer that for all x_8, if x_8 writes a letter, then x_8 is a nurse. From the fact that for all x_8, if x_8 were a nurse, then x_8 is playing a game, and that there is at least one x_8 for which if x_8 plays a game, then x_8 is writing a letter we can infer that there is at least one x_8 for which if x_8 were a nurse, then x_8 is writing a letter. Finally, from the fact that for all x_8, if x_8 writes a letter, then x_8 is a nurse, and that there is at least one x_8 for which if x_8 were a nurse, then x_8 is writing a letter we can infer that there is at least one x_8 for which x_8 will write a letter if and only if x_8 is a nurse. Therefore, the answer is yes.