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