Consider the following premises. For all x_17, if x_17 cooks, then x_17 is an electrician. If Joseph were an electrician, then David is not running. It is raining. For all x_17, if x_17 were an electrician, then x_17 is cooking. If it rains, then David is running. Can we infer the following from them? If we can, name the inference rule being used: Joseph is not cooking.
 From the fact that for all x_17, if x_17 cooks, then x_17 is an electrician, and that for all x_17, if x_17 were an electrician, then x_17 is cooking we can infer that for all x_17, x_17 will cook if and only if x_17 is an electrician via universal biconditional introduction. From the fact that if it rains, then David is running, and that it is raining we can infer that David is running via modus ponens. From the fact that for all x_17, x_17 will cook if and only if x_17 is an electrician we can infer that Joseph is cooking if and only if Joseph is an electrician via universal instantiation. From the fact that if Joseph were an electrician, then David is not running, and that David is running we can infer that Joseph is not an electrician via modus tollens. Finally, from the fact that Joseph is cooking if and only if Joseph is an electrician, and that Joseph is not an electrician we can infer that Joseph is not cooking via biconditional elimination. Therefore, the answer is yes.