Consider the following premises. David is a doctor or John is not writing a letter. If David were a doctor, then John is writing a letter. If John writes a letter, then David is a doctor. Can we infer the following from them? If is not true that John is writing a letter if and only if David is a doctor.
 From the fact that if David were a doctor, then John is writing a letter, and that if John writes a letter, then David is a doctor we can infer that David is a doctor if and only if John is writing a letter. From the fact that David is a doctor or John is not writing a letter we can infer that if John writes a letter, then David is a doctor. From the fact that David is a doctor if and only if John is writing a letter we can infer that if David were a doctor, then John is writing a letter. Finally, from the fact that if John writes a letter, then David is a doctor, and that if David were a doctor, then John is writing a letter we can infer that John is writing a letter if and only if David is a doctor, which contradicts that If is not true that John is writing a letter if and only if David is a doctor. Therefore, the answer is no.