Consider the following premises. For all x_6, x_6 will cook. For all x_6, x_6 is a nurse or x_6 will not cook. There is at least one x_6 for which x_6 is a doctor. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, either x_6 is a doctor or x_6 is a nurse are not true.
 From the fact that for all x_6, x_6 is a nurse or x_6 will not cook we can infer that for all x_6, if x_6 cooks, then x_6 is a nurse via universal rewriting. From the fact that for all x_6, if x_6 cooks, then x_6 is a nurse, and that for all x_6, x_6 will cook we can infer that for all x_6, x_6 is a nurse via universal modus ponens. Finally, from the fact that there is at least one x_6 for which x_6 is a doctor, and that for all x_6, x_6 is a nurse we can infer that some doctors are nurses via existential conjunction, which contradicts that For all x_6, either x_6 is a doctor or x_6 is a nurse are not true. Therefore, the answer is no.