Consider the following premises. For all x_1, if x_1 were a politician, then x_1 is writing a letter. Sarah is a politician if and only if Sarah is writing a letter. Can we infer the following from them? Sarah is writing a letter if and only if Sarah is a politician.
 From the fact that for all x_1, if x_1 were a politician, then x_1 is writing a letter we can infer that if Sarah were a politician, then Sarah is writing a letter. From the fact that Sarah is a politician if and only if Sarah is writing a letter we can infer that if Sarah writes a letter, then Sarah is a politician. Finally, from the fact that if Sarah writes a letter, then Sarah is a politician, and that if Sarah were a politician, then Sarah is writing a letter we can infer that Sarah is writing a letter if and only if Sarah is a politician. Therefore, the answer is yes.