Consider the following premises. 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 bored. For all x_2, if x_2 were bored, then x_2 is an artist. For all x_2, if x_2 were an artist, then x_2 is cooking. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_2 for which x_2 is not an artist.
 From the fact that for all x_2, if x_2 cooks, then x_2 is bored, and that for all x_2, if x_2 were bored, then x_2 is an artist we can infer that for all x_2, if x_2 cooks, then x_2 is an artist via universal transitivity. From the fact that for all x_2, if x_2 were an artist, then x_2 is cooking, and that for all x_2, if x_2 cooks, then x_2 is an artist we can infer that for all x_2, x_2 is an artist if and only if x_2 will cook via universal biconditional introduction. Finally, from the fact that for all x_2, x_2 is an artist if and only if x_2 will cook, 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 an artist via existential biconditional elimination. Therefore, the answer is yes.