Consider the following premises. For all x_8, if x_8 were a doctor, then x_8 is playing squash. For all x_8, x_8 will sleep and x_8 will cook. For all x_8, if x_8 were a mechanic, then x_8 is playing squash. For all x_8, x_8 is a doctor or x_8 is a mechanic. 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 either x_8 will play squash or x_8 will sleep are not true.
 From the fact that for all x_8, if x_8 were a doctor, then x_8 is playing squash, that for all x_8, if x_8 were a mechanic, then x_8 is playing squash, and that for all x_8, x_8 is a doctor or x_8 is a mechanic we can infer that for all x_8, x_8 will play squash via universal disjunction elimination. From the fact that for all x_8, x_8 will sleep and x_8 will cook we can infer that for all x_8, x_8 will sleep via universal simplification. Finally, from the fact that for all x_8, x_8 will play squash, and that for all x_8, x_8 will sleep we can infer that for all x_8, x_8 will play squash and x_8 will sleep via universal conjunction, which contradicts that There is at least one x_8 for which either x_8 will play squash or x_8 will sleep are not true. Therefore, the answer is no.