Consider the following premises. For all z, if z were a politician, then z is writing a letter. For all z, if z writes a letter, then z is a politician. There is at least one z for which if z were a politician, then z is writing a letter. Can we infer the following from them? For all z, the fact that z were a politician does not imply that z is writing a letter.
That contradicts the premises. Therefore the answer is no.