Consider the following premises. For all x_2, x_2 will read a book. For all x_2, x_2 will listen to a song. There is at least one x_2 for which if x_2 listens to a song, then x_2 is playing a game. Can we infer the following from them? If we can, name the inference rule being used: For all x_2, either x_2 will read a book or x_2 will play a game are not true.
 From the fact that there is at least one x_2 for which if x_2 listens to a song, then x_2 is playing a game, and that for all x_2, x_2 will listen to a song we can infer that there is at least one x_2 for which x_2 will play a game via existential modus ponens. Finally, from the fact that for all x_2, x_2 will read a book, and that there is at least one x_2 for which x_2 will play a game we can infer that there is at least one x_2 for which x_2 will read a book and x_2 will play a game via existential conjunction, which contradicts that For all x_2, either x_2 will read a book or x_2 will play a game are not true. Therefore, the answer is no.