Consider the following premises. For all x_3, if x_3 were curious, then x_3 is running. For all x_3, x_3 is curious. There is at least one x_3 for which x_3 will not go running. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_3 for which x_3 will play tennis.
 From the fact that for all x_3, if x_3 were curious, then x_3 is running, and that for all x_3, x_3 is curious we can infer that for all x_3, x_3 will go running via universal modus ponens. From the fact that for all x_3, x_3 will go running we can infer that for all x_3, x_3 will go running or x_3 will play tennis via universal addition. Finally, from the fact that for all x_3, x_3 will go running or x_3 will play tennis, and that there is at least one x_3 for which x_3 will not go running we can infer that there is at least one x_3 for which x_3 will play tennis via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.