Consider the following premises. For all x_6, x_6 will play a game. For all x_6, if x_6 goes running, then x_6 is drinking water. For all x_6, x_6 is happy. For all x_6, x_6 will go running. 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 drink water or x_6 will play a game are not true.
 From the fact that for all x_6, if x_6 goes running, then x_6 is drinking water, and that for all x_6, x_6 will go running we can infer that for all x_6, x_6 will drink water via universal modus ponens. From the fact that for all x_6, x_6 will drink water, and that for all x_6, x_6 is happy we can infer that for all x_6, x_6 will drink water and x_6 is happy via universal conjunction. From the fact that for all x_6, x_6 will drink water and x_6 is happy we can infer that for all x_6, x_6 will drink water via universal simplification. Finally, from the fact that for all x_6, x_6 will drink water, and that for all x_6, x_6 will play a game we can infer that for all x_6, x_6 will drink water and x_6 will play a game via universal conjunction, which contradicts that There is at least one x_6 for which either x_6 will drink water or x_6 will play a game are not true. Therefore, the answer is no.