Consider the following premises. If Thomas plays tennis, then Thomas is rich. For all x_4, x_4 is rich if and only if x_4 will play tennis. For all x_4, if x_4 were rich, then x_4 is playing tennis. Can we infer the following from them? If we can, name the inference rule being used: Thomas is rich if and only if Thomas is playing tennis.
 From the fact that for all x_4, x_4 is rich if and only if x_4 will play tennis we can infer that for all x_4, if x_4 plays tennis, then x_4 is rich via universal biconditional elimination. From the fact that for all x_4, if x_4 were rich, then x_4 is playing tennis, and that for all x_4, if x_4 plays tennis, then x_4 is rich we can infer that for all x_4, x_4 is rich if and only if x_4 will play tennis via universal biconditional introduction. Finally, from the fact that for all x_4, x_4 is rich if and only if x_4 will play tennis we can infer that Thomas is rich if and only if Thomas is playing tennis via universal instantiation. Therefore, the answer is yes.