Consider the following premises. There is at least one y_2 for which y_2 will read a book. For all y_2, if y_2 reads a book, then y_2 is cooking. For all y_2, y_2 will read a book or y_2 will not cook. Can we infer the following from them? If we can, name the inference rule being used: For all y_2, y_2 will not cook.
 From the fact that for all y_2, y_2 will read a book or y_2 will not cook we can infer that for all y_2, if y_2 cooks, then y_2 is reading a book via universal rewriting. From the fact that for all y_2, if y_2 cooks, then y_2 is reading a book, and that for all y_2, if y_2 reads a book, then y_2 is cooking we can infer that for all y_2, y_2 will cook if and only if y_2 will read a book via universal biconditional introduction. From the fact that for all y_2, y_2 will cook if and only if y_2 will read a book we can infer that for all y_2, if y_2 reads a book, then y_2 is cooking via universal biconditional elimination. Finally, from the fact that for all y_2, if y_2 reads a book, then y_2 is cooking, and that there is at least one y_2 for which y_2 will read a book we can infer that there is at least one y_2 for which y_2 will cook via existential modus ponens, which contradicts that For all y_2, y_2 will not cook. Therefore, the answer is no.