Consider the following premises. If Michael were happy, then John is not a nurse. John is a nurse. Michael is happy if and only if Karen is cooking. If Michael were happy, then Karen is cooking. Can we infer the following from them? If we can, name the inference rule being used: Karen is not cooking or Joseph is an electrician.
 From the fact that Michael is happy if and only if Karen is cooking we can infer that if Karen cooks, then Michael is happy via biconditional elimination. From the fact that if Karen cooks, then Michael is happy, and that if Michael were happy, then Karen is cooking we can infer that Karen is cooking if and only if Michael is happy via biconditional introduction. From the fact that if Michael were happy, then John is not a nurse, and that John is a nurse we can infer that Michael is not happy via modus tollens. From the fact that Karen is cooking if and only if Michael is happy, and that Michael is not happy we can infer that Karen is not cooking via biconditional elimination. Finally, from the fact that Karen is not cooking we can infer that Karen is not cooking or Joseph is an electrician via addition. Therefore, the answer is yes.