Consider the following premises. For all x_3, x_3 is a scientist if and only if x_3 will write a letter. For all x_3, x_3 will not write a letter. There is at least one x_3 for which if x_3 writes a letter, then x_3 is a scientist. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_3 for which x_3 is not a scientist.
 From the fact that for all x_3, x_3 is a scientist if and only if x_3 will write a letter we can infer that for all x_3, if x_3 were a scientist, then x_3 is writing a letter via universal biconditional elimination. From the fact that for all x_3, if x_3 were a scientist, then x_3 is writing a letter, and that there is at least one x_3 for which if x_3 writes a letter, then x_3 is a scientist we can infer that there is at least one x_3 for which x_3 is a scientist if and only if x_3 will write a letter via existential biconditional introduction. Finally, from the fact that there is at least one x_3 for which x_3 is a scientist if and only if x_3 will write a letter, and that for all x_3, x_3 will not write a letter we can infer that there is at least one x_3 for which x_3 is not a scientist via existential biconditional elimination. Therefore, the answer is yes.