Consider the following premises. For all x, x will climb a mountain or x will drink water. There is at least one x for which x will not climb a mountain. For all x, if x plays tennis, then x is not drinking water. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which x will not play tennis.
 From the fact that for all x, x will climb a mountain or x will drink water, and that there is at least one x for which x will not climb a mountain we can infer that there is at least one x for which x will drink water via existential disjunctive syllogism. Finally, from the fact that for all x, if x plays tennis, then x is not drinking water, and that there is at least one x for which x will drink water we can infer that there is at least one x for which x will not play tennis via existential modus tollens. Therefore, the answer is yes.