Consider the following premises. For all x_5, if x_5 drives a car, then x_5 is rich. There is at least one x_5 for which if x_5 were a doctor, then x_5 is driving a car. For all x_5, if x_5 were rich, then x_5 is a scientist. For all x_5, if x_5 were a scientist, then x_5 is a doctor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which x_5 will drive a car if and only if x_5 is a doctor.
 From the fact that for all x_5, if x_5 were rich, then x_5 is a scientist, and that for all x_5, if x_5 were a scientist, then x_5 is a doctor we can infer that for all x_5, if x_5 were rich, then x_5 is a doctor via universal transitivity. From the fact that for all x_5, if x_5 drives a car, then x_5 is rich, and that for all x_5, if x_5 were rich, then x_5 is a doctor we can infer that for all x_5, if x_5 drives a car, then x_5 is a doctor via universal transitivity. Finally, from the fact that for all x_5, if x_5 drives a car, then x_5 is a doctor, and that there is at least one x_5 for which if x_5 were a doctor, then x_5 is driving a car we can infer that there is at least one x_5 for which x_5 will drive a car if and only if x_5 is a doctor via existential biconditional introduction. Therefore, the answer is yes.