Consider the following premises. If Barbara were a doctor, then Jennifer is playing squash. Barbara is a soldier if and only if David is listening to a song. If Jennifer plays squash, then David is listening to a song. If Barbara were a soldier, then Barbara is a doctor. Can we infer the following from them? If we can, name the inference rule being used: If is not true that Jennifer is playing squash if and only if Barbara is a soldier.
 From the fact that Barbara is a soldier if and only if David is listening to a song we can infer that if David listens to a song, then Barbara is a soldier via biconditional elimination. From the fact that if Barbara were a soldier, then Barbara is a doctor, and that if Barbara were a doctor, then Jennifer is playing squash we can infer that if Barbara were a soldier, then Jennifer is playing squash via transitivity. From the fact that if Jennifer plays squash, then David is listening to a song, and that if David listens to a song, then Barbara is a soldier we can infer that if Jennifer plays squash, then Barbara is a soldier via transitivity. Finally, from the fact that if Jennifer plays squash, then Barbara is a soldier, and that if Barbara were a soldier, then Jennifer is playing squash we can infer that Jennifer is playing squash if and only if Barbara is a soldier via biconditional introduction, which contradicts that If is not true that Jennifer is playing squash if and only if Barbara is a soldier. Therefore, the answer is no.