Consider the following premises. For all x_8, x_8 is a writer if and only if x_8 is a lawyer. There is at least one x_8 for which if x_8 writes a letter, then x_8 is poor. For all x_8, if x_8 were poor, then x_8 is a writer. 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 x_8 writes a letter, then x_8 is a lawyer.
 From the fact that for all x_8, x_8 is a writer if and only if x_8 is a lawyer we can infer that for all x_8, if x_8 were a writer, then x_8 is a lawyer via universal biconditional elimination. From the fact that there is at least one x_8 for which if x_8 writes a letter, then x_8 is poor, and that for all x_8, if x_8 were poor, then x_8 is a writer we can infer that there is at least one x_8 for which if x_8 writes a letter, then x_8 is a writer via existential transitivity. Finally, from the fact that there is at least one x_8 for which if x_8 writes a letter, then x_8 is a writer, and that for all x_8, if x_8 were a writer, then x_8 is a lawyer we can infer that there is at least one x_8 for which if x_8 writes a letter, then x_8 is a lawyer via existential transitivity. Therefore, the answer is yes.