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