Consider the following premises. There is at least one x_6 for which if x_6 plays tennis, then x_6 is running. For all x_6, if x_6 were a doctor, then x_6 is playing tennis. For all x_6, x_6 is a doctor or x_6 will not go running. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which x_6 will go running if and only if x_6 will play tennis.
 From the fact that for all x_6, x_6 is a doctor or x_6 will not go running we can infer that for all x_6, if x_6 goes running, then x_6 is a doctor via universal rewriting. From the fact that for all x_6, if x_6 goes running, then x_6 is a doctor, and that for all x_6, if x_6 were a doctor, then x_6 is playing tennis we can infer that for all x_6, if x_6 goes running, then x_6 is playing tennis via universal transitivity. Finally, from the fact that for all x_6, if x_6 goes running, then x_6 is playing tennis, and that there is at least one x_6 for which if x_6 plays tennis, then x_6 is running we can infer that there is at least one x_6 for which x_6 will go running if and only if x_6 will play tennis via existential biconditional introduction. Therefore, the answer is yes.