Consider the following premises. There is at least one x_5 for which x_5 will play a game. For all x_5, x_5 will read a book. For all x_5, if x_5 plays a game, then x_5 is not reading a book. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which x_5 is sad.
 From the fact that for all x_5, if x_5 plays a game, then x_5 is not reading a book, and that for all x_5, x_5 will read a book we can infer that for all x_5, x_5 will not play a game via universal modus tollens. From the fact that there is at least one x_5 for which x_5 will play a game we can infer that there is at least one x_5 for which x_5 will play a game or x_5 is sad via existential addition. Finally, from the fact that there is at least one x_5 for which x_5 will play a game or x_5 is sad, and that for all x_5, x_5 will not play a game we can infer that there is at least one x_5 for which x_5 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.