Consider the following premises. There is at least one z_4 for which if z_4 goes running, then z_4 is cooking. For all z_4, z_4 will read a book or z_4 will go running. For all z_4, if z_4 reads a book, then z_4 is cooking. For all z_4, if z_4 cooks, then z_4 is reading a book. Can we infer the following from them? If we can, name the inference rule being used: For all z_4, z_4 will not cook.
 From the fact that for all z_4, if z_4 cooks, then z_4 is reading a book, and that for all z_4, if z_4 reads a book, then z_4 is cooking we can infer that for all z_4, z_4 will cook if and only if z_4 will read a book via universal biconditional introduction. From the fact that for all z_4, z_4 will cook if and only if z_4 will read a book we can infer that for all z_4, if z_4 reads a book, then z_4 is cooking via universal biconditional elimination. Finally, from the fact that for all z_4, if z_4 reads a book, then z_4 is cooking, that there is at least one z_4 for which if z_4 goes running, then z_4 is cooking, and that for all z_4, z_4 will read a book or z_4 will go running we can infer that there is at least one z_4 for which z_4 will cook via existential disjunction elimination, which contradicts that For all z_4, z_4 will not cook. Therefore, the answer is no.