Consider the following premises. For all z, if z writes a letter, then z is a doctor. For all z, z is not a doctor. There is at least one z for which z will write a letter if and only if z is happy. For all z, if z were a doctor, then z is writing a letter. Can we infer the following from them? For all z, z is happy.
 From the fact that for all z, if z writes a letter, then z is a doctor, and that for all z, if z were a doctor, then z is writing a letter we can infer that for all z, z will write a letter if and only if z is a doctor. From the fact that for all z, z will write a letter if and only if z is a doctor, and that for all z, z is not a doctor we can infer that for all z, z will not write a letter. Finally, from the fact that there is at least one z for which z will write a letter if and only if z is happy, and that for all z, z will not write a letter we can infer that there is at least one z for which z is not happy, which contradicts that For all z, z is happy. Therefore, the answer is no.