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