Consider the following premises. For all x_2, x_2 is a doctor. For all x_2, x_2 is not an artist. For all x_2, x_2 is an artist or x_2 will cook. 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 either x_2 is a doctor or x_2 will cook are not true.
 From the fact that for all x_2, x_2 is an artist or x_2 will cook, and that for all x_2, x_2 is not an artist we can infer that for all x_2, x_2 will cook via universal disjunctive syllogism. Finally, from the fact that for all x_2, x_2 is a doctor, and that for all x_2, x_2 will cook we can infer that for all x_2, x_2 is a doctor and x_2 will cook via universal conjunction, which contradicts that There is at least one x_2 for which either x_2 is a doctor or x_2 will cook are not true. Therefore, the answer is no.