Consider the following premises. For all x_5, if x_5 writes a letter, then x_5 is a scientist. For all x_5, x_5 will write a letter if and only if x_5 will go running. For all x_5, if x_5 were a scientist, then x_5 is running. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which if is not true that x_5 will write a letter if and only if x_5 will go running.
 From the fact that for all x_5, if x_5 writes a letter, then x_5 is a scientist, and that for all x_5, if x_5 were a scientist, then x_5 is running we can infer that for all x_5, if x_5 writes a letter, then x_5 is running via universal transitivity. From the fact that for all x_5, x_5 will write a letter if and only if x_5 will go running we can infer that for all x_5, if x_5 goes running, then x_5 is writing a letter via universal biconditional elimination. Finally, from the fact that for all x_5, if x_5 writes a letter, then x_5 is running, and that for all x_5, if x_5 goes running, then x_5 is writing a letter we can infer that for all x_5, x_5 will write a letter if and only if x_5 will go running via universal biconditional introduction, which contradicts that There is at least one x_5 for which if is not true that x_5 will write a letter if and only if x_5 will go running. Therefore, the answer is no.