Consider the following premises. For all x_6, if x_6 plays a game, then x_6 is working. For all x_6, x_6 will sleep. For all x_6, x_6 will cook. For all x_6, x_6 will play a game. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which either x_6 will cook or x_6 will sleep are not true.
 From the fact that for all x_6, if x_6 plays a game, then x_6 is working, and that for all x_6, x_6 will play a game we can infer that for all x_6, x_6 will work via universal modus ponens. From the fact that for all x_6, x_6 will cook, and that for all x_6, x_6 will work we can infer that for all x_6, x_6 will cook and x_6 will work via universal conjunction. From the fact that for all x_6, x_6 will cook and x_6 will work we can infer that for all x_6, x_6 will cook via universal simplification. Finally, from the fact that for all x_6, x_6 will cook, and that for all x_6, x_6 will sleep we can infer that for all x_6, x_6 will cook and x_6 will sleep via universal conjunction, which contradicts that There is at least one x_6 for which either x_6 will cook or x_6 will sleep are not true. Therefore, the answer is no.