Consider the following premises. For all x_7, x_7 will not drink water. For all x_7, x_7 will drink water. There is at least one x_7 for which x_7 will not play squash. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_7 for which x_7 is sad.
 From the fact that for all x_7, x_7 will drink water we can infer that for all x_7, x_7 will drink water or x_7 will play squash via universal addition. From the fact that for all x_7, x_7 will drink water or x_7 will play squash, and that for all x_7, x_7 will not drink water we can infer that for all x_7, x_7 will play squash via universal disjunctive syllogism. From the fact that for all x_7, x_7 will play squash we can infer that for all x_7, x_7 will play squash or x_7 is sad via universal addition. Finally, from the fact that for all x_7, x_7 will play squash or x_7 is sad, and that there is at least one x_7 for which x_7 will not play squash we can infer that there is at least one x_7 for which x_7 is sad via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.