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