Consider the following premises. For all x_6, if x_6 writes a letter, then x_6 is making tea. For all x_6, x_6 is poor if and only if x_6 will write a letter. There is at least one x_6 for which if x_6 makes tea, then x_6 is poor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which x_6 will write a letter if and only if x_6 is poor.
 From the fact that for all x_6, if x_6 writes a letter, then x_6 is making tea, and that there is at least one x_6 for which if x_6 makes tea, then x_6 is poor we can infer that there is at least one x_6 for which if x_6 writes a letter, then x_6 is poor via existential transitivity. From the fact that for all x_6, x_6 is poor if and only if x_6 will write a letter we can infer that for all x_6, if x_6 were poor, then x_6 is writing a letter via universal biconditional elimination. Finally, from the fact that there is at least one x_6 for which if x_6 writes a letter, then x_6 is poor, and that for all x_6, if x_6 were poor, then x_6 is writing a letter we can infer that there is at least one x_6 for which x_6 will write a letter if and only if x_6 is poor via existential biconditional introduction. Therefore, the answer is yes.