Consider the following premises. Joseph is sad or James is not cooking. William is not listening to a song. William is listening to a song if and only if Joseph is sad. If Joseph were sad, then James is cooking. Can we infer the following from them? If we can, name the inference rule being used: James is not cooking.
 From the fact that William is listening to a song if and only if Joseph is sad, and that William is not listening to a song we can infer that Joseph is not sad via biconditional elimination. From the fact that Joseph is sad or James is not cooking we can infer that if James cooks, then Joseph is sad via rewriting. From the fact that if James cooks, then Joseph is sad, and that if Joseph were sad, then James is cooking we can infer that James is cooking if and only if Joseph is sad via biconditional introduction. Finally, from the fact that James is cooking if and only if Joseph is sad, and that Joseph is not sad we can infer that James is not cooking via biconditional elimination. Therefore, the answer is yes.