Consider the following premises. For all x_3, x_3 will read a book or x_3 will play a game. For all x_3, x_3 will play squash. For all x_3, x_3 will play a game if and only if x_3 is excited. For all x_3, x_3 will not read a book or x_3 will listen to a song. For all x_3, x_3 is not excited. 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 either x_3 will listen to a song or x_3 will play squash are not true.
 From the fact that for all x_3, x_3 will read a book or x_3 will play a game, and that for all x_3, x_3 will not read a book or x_3 will listen to a song we can infer that for all x_3, x_3 will play a game or x_3 will listen to a song via universal resolution. From the fact that for all x_3, x_3 will play a game if and only if x_3 is excited, and that for all x_3, x_3 is not excited we can infer that for all x_3, x_3 will not play a game via universal biconditional elimination. From the fact that for all x_3, x_3 will play a game or x_3 will listen to a song, and that for all x_3, x_3 will not play a game we can infer that for all x_3, x_3 will listen to a song via universal disjunctive syllogism. Finally, from the fact that for all x_3, x_3 will listen to a song, and that for all x_3, x_3 will play squash we can infer that for all x_3, x_3 will listen to a song and x_3 will play squash via universal conjunction, which contradicts that There is at least one x_3 for which either x_3 will listen to a song or x_3 will play squash are not true. Therefore, the answer is no.