Consider the following premises. For all z_4, z_4 will not write a letter. For all z_4, z_4 will play a game if and only if z_4 will read a book. For all z_4, z_4 will write a letter if and only if z_4 will play a game. For all z_4, if z_4 reads a book, then z_4 is playing a game. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z_4 for which z_4 will read a book.
 From the fact that for all z_4, z_4 will play a game if and only if z_4 will read a book we can infer that for all z_4, if z_4 plays a game, then z_4 is reading a book via universal biconditional elimination. From the fact that for all z_4, z_4 will write a letter if and only if z_4 will play a game, and that for all z_4, z_4 will not write a letter we can infer that for all z_4, z_4 will not play a game via universal biconditional elimination. From the fact that for all z_4, if z_4 reads a book, then z_4 is playing a game, and that for all z_4, if z_4 plays a game, then z_4 is reading a book we can infer that for all z_4, z_4 will read a book if and only if z_4 will play a game via universal biconditional introduction. Finally, from the fact that for all z_4, z_4 will read a book if and only if z_4 will play a game, and that for all z_4, z_4 will not play a game we can infer that for all z_4, z_4 will not read a book via universal biconditional elimination, which contradicts that There is at least one z_4 for which z_4 will read a book. Therefore, the answer is no.