Consider the following premises. For all x_8, if x_8 writes a letter, then x_8 is bored. For all x_8, x_8 will write a letter if and only if x_8 will read a book. For all x_8, if x_8 were bored, then x_8 is reading a book. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_8 for which if is not true that x_8 is bored if and only if x_8 will write a letter.
 From the fact that for all x_8, x_8 will write a letter if and only if x_8 will read a book we can infer that for all x_8, if x_8 reads a book, then x_8 is writing a letter via universal biconditional elimination. From the fact that for all x_8, if x_8 were bored, then x_8 is reading a book, and that for all x_8, if x_8 reads a book, then x_8 is writing a letter we can infer that for all x_8, if x_8 were bored, then x_8 is writing a letter via universal transitivity. Finally, from the fact that for all x_8, if x_8 were bored, then x_8 is writing a letter, and that for all x_8, if x_8 writes a letter, then x_8 is bored we can infer that for all x_8, x_8 is bored if and only if x_8 will write a letter via universal biconditional introduction, which contradicts that There is at least one x_8 for which if is not true that x_8 is bored if and only if x_8 will write a letter. Therefore, the answer is no.