Consider the following premises. For all x_2, if x_2 were a writer, then x_2 is cooking. There is at least one x_2 for which x_2 will not cook. For all x_2, if x_2 cooks, then x_2 is a writer. For all x_2, x_2 is a writer or x_2 will not cook. Can we infer the following from them? If we can, name the inference rule being used: For all x_2, x_2 is a writer.
 From the fact that for all x_2, x_2 is a writer or x_2 will not cook we can infer that for all x_2, if x_2 cooks, then x_2 is a writer via universal rewriting. From the fact that for all x_2, if x_2 cooks, then x_2 is a writer, and that for all x_2, if x_2 were a writer, then x_2 is cooking we can infer that for all x_2, x_2 will cook if and only if x_2 is a writer via universal biconditional introduction. From the fact that for all x_2, x_2 will cook if and only if x_2 is a writer we can infer that for all x_2, if x_2 were a writer, then x_2 is cooking via universal biconditional elimination. From the fact that for all x_2, if x_2 cooks, then x_2 is a writer, and that for all x_2, if x_2 were a writer, then x_2 is cooking we can infer that for all x_2, x_2 will cook if and only if x_2 is a writer via universal biconditional introduction. Finally, from the fact that for all x_2, x_2 will cook if and only if x_2 is a writer, and that there is at least one x_2 for which x_2 will not cook we can infer that there is at least one x_2 for which x_2 is not a writer via existential biconditional elimination, which contradicts that For all x_2, x_2 is a writer. Therefore, the answer is no.