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