Consider the following premises. For all x_2, x_2 will read a book if and only if x_2 will go running. For all x_2, if x_2 goes running, then x_2 is reading a book. For all x_2, if x_2 reads a book, then x_2 is running. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_2 for which if is not true that x_2 will read a book if and only if x_2 will go running.
 From the fact that for all x_2, x_2 will read a book if and only if x_2 will go running we can infer that for all x_2, if x_2 reads a book, then x_2 is running via universal biconditional elimination. From the fact that for all x_2, if x_2 reads a book, then x_2 is running, and that for all x_2, if x_2 goes running, then x_2 is reading a book we can infer that for all x_2, x_2 will read a book if and only if x_2 will go running via universal biconditional introduction. From the fact that for all x_2, x_2 will read a book if and only if x_2 will go running we can infer that for all x_2, if x_2 goes running, then x_2 is reading a book via universal biconditional elimination. Finally, from the fact that for all x_2, if x_2 reads a book, then x_2 is running, and that for all x_2, if x_2 goes running, then x_2 is reading a book we can infer that for all x_2, x_2 will read a book if and only if x_2 will go running via universal biconditional introduction, which contradicts that There is at least one x_2 for which if is not true that x_2 will read a book if and only if x_2 will go running. Therefore, the answer is no.