Consider the following premises. For all x_8, x_8 is a mechanic. For all x_8, x_8 will cook or x_8 is not a mechanic. For all x_8, x_8 will cook or x_8 is not a nurse. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_8 for which x_8 will not cook.
 From the fact that for all x_8, x_8 will cook or x_8 is not a mechanic we can infer that for all x_8, if x_8 were a mechanic, then x_8 is cooking via universal rewriting. From the fact that for all x_8, x_8 will cook or x_8 is not a nurse we can infer that for all x_8, if x_8 were a nurse, then x_8 is cooking via universal rewriting. From the fact that for all x_8, x_8 is a mechanic we can infer that for all x_8, x_8 is a mechanic or x_8 is a nurse via universal addition. Finally, from the fact that for all x_8, if x_8 were a mechanic, then x_8 is cooking, that for all x_8, if x_8 were a nurse, then x_8 is cooking, and that for all x_8, x_8 is a mechanic or x_8 is a nurse we can infer that for all x_8, x_8 will cook via universal disjunction elimination, which contradicts that There is at least one x_8 for which x_8 will not cook. Therefore, the answer is no.