Consider the following premises. For all x_6, if x_6 writes a letter, then x_6 is playing a game. For all x_6, if x_6 reads a book, then x_6 is writing a letter. For all x_6, if x_6 plays a game, then x_6 is reading a book. For all x_6, x_6 will not read a book. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 will not write a letter.
 From the fact that for all x_6, if x_6 writes a letter, then x_6 is playing a game, and that for all x_6, if x_6 plays a game, then x_6 is reading a book we can infer that for all x_6, if x_6 writes a letter, then x_6 is reading a book via universal transitivity. From the fact that for all x_6, if x_6 writes a letter, then x_6 is reading a book, and that for all x_6, if x_6 reads a book, then x_6 is writing a letter we can infer that for all x_6, x_6 will write a letter if and only if x_6 will read a book via universal biconditional introduction. From the fact that for all x_6, x_6 will write a letter if and only if x_6 will read a book we can infer that for all x_6, if x_6 writes a letter, then x_6 is reading a book via universal biconditional elimination. Finally, from the fact that for all x_6, if x_6 writes a letter, then x_6 is reading a book, and that for all x_6, x_6 will not read a book we can infer that for all x_6, x_6 will not write a letter via universal modus tollens. Therefore, the answer is yes.