Consider the following premises. For all x_3, if x_3 were a writer, then x_3 is cooking. For all x_3, if x_3 plays a game, then x_3 is a writer. For all x_3, x_3 will play a game. For all x_3, if x_3 were fast, then x_3 is cooking. 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 not cook.
 From the fact that for all x_3, if x_3 plays a game, then x_3 is a writer, and that for all x_3, x_3 will play a game we can infer that for all x_3, x_3 is a writer via universal modus ponens. From the fact that for all x_3, x_3 is a writer we can infer that for all x_3, x_3 is a writer or x_3 is fast via universal addition. Finally, from the fact that for all x_3, if x_3 were a writer, then x_3 is cooking, that for all x_3, if x_3 were fast, then x_3 is cooking, and that for all x_3, x_3 is a writer or x_3 is fast we can infer that for all x_3, x_3 will cook via universal disjunction elimination, which contradicts that There is at least one x_3 for which x_3 will not cook. Therefore, the answer is no.