Consider the following premises. There is at least one z_3 for which z_3 is a politician if and only if z_3 is a doctor. For all z_3, z_3 is a politician or z_3 is not a doctor. Can we infer the following from them? There is at least one z_3 for which z_3 is a doctor if and only if z_3 is a politician.
 From the fact that for all z_3, z_3 is a politician or z_3 is not a doctor we can infer that for all z_3, if z_3 were a doctor, then z_3 is a politician. From the fact that there is at least one z_3 for which z_3 is a politician if and only if z_3 is a doctor we can infer that there is at least one z_3 for which if z_3 were a politician, then z_3 is a doctor. Finally, from the fact that for all z_3, if z_3 were a doctor, then z_3 is a politician, and that there is at least one z_3 for which if z_3 were a politician, then z_3 is a doctor we can infer that there is at least one z_3 for which z_3 is a doctor if and only if z_3 is a politician. Therefore, the answer is yes.