Consider the following premises. For all x_6, x_6 will not drink water or x_6 will go running. For all x_6, x_6 is not sad. For all x_6, x_6 will drink water or x_6 is sad. For all x_6, x_6 will climb a mountain. 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 go running or x_6 will climb a mountain are not true.
 From the fact that for all x_6, x_6 will drink water or x_6 is sad, and that for all x_6, x_6 will not drink water or x_6 will go running we can infer that for all x_6, x_6 is sad or x_6 will go running via universal resolution. From the fact that for all x_6, x_6 is sad or x_6 will go running, and that for all x_6, x_6 is not sad we can infer that for all x_6, x_6 will go running via universal disjunctive syllogism. Finally, from the fact that for all x_6, x_6 will go running, and that for all x_6, x_6 will climb a mountain we can infer that for all x_6, x_6 will go running and x_6 will climb a mountain via universal conjunction, which contradicts that There is at least one x_6 for which either x_6 will go running or x_6 will climb a mountain are not true. Therefore, the answer is no.