Consider the following premises. If it is cold, then Elisabeth is sleeping. It is late if and only if Joseph is a musician. It is not late. Joseph is a musician if and only if Elisabeth is sleeping. Can we infer the following from them? If we can, name the inference rule being used: It is not cold.
 From the fact that it is late if and only if Joseph is a musician we can infer that if Joseph were a musician, then it is late via biconditional elimination. From the fact that Joseph is a musician if and only if Elisabeth is sleeping we can infer that if Elisabeth sleeps, then Joseph is a musician via biconditional elimination. From the fact that if it is cold, then Elisabeth is sleeping, and that if Elisabeth sleeps, then Joseph is a musician we can infer that if it is cold, then Joseph is a musician via transitivity. From the fact that if it is cold, then Joseph is a musician, and that if Joseph were a musician, then it is late we can infer that if it is cold, then it is late via transitivity. Finally, from the fact that if it is cold, then it is late, and that it is not late we can infer that it is not cold via modus tollens. Therefore, the answer is yes.