Consider the following premises. James is bored if and only if Elisabeth is happy. Richard is not a nurse. If James were bored, then Richard is a nurse. Richard is a musician if and only if Elisabeth is happy. Can we infer the following from them? If we can, name the inference rule being used: Richard is not a musician.
 From the fact that James is bored if and only if Elisabeth is happy we can infer that if Elisabeth were happy, then James is bored via biconditional elimination. From the fact that if Elisabeth were happy, then James is bored, and that if James were bored, then Richard is a nurse we can infer that if Elisabeth were happy, then Richard is a nurse via transitivity. From the fact that if Elisabeth were happy, then Richard is a nurse, and that Richard is not a nurse we can infer that Elisabeth is not happy via modus tollens. Finally, from the fact that Richard is a musician if and only if Elisabeth is happy, and that Elisabeth is not happy we can infer that Richard is not a musician via biconditional elimination. Therefore, the answer is yes.