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