Consider the following premises. For all x_10, if x_10 were poor, then x_10 is an electrician. There is at least one x_10 for which x_10 will not write a letter. For all x_10, if x_10 were an electrician, then x_10 is poor. For all x_10, if x_10 writes a letter, then x_10 is poor. For all x_10, if x_10 were poor, then x_10 is writing a letter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which x_10 is not an electrician.
 From the fact that for all x_10, if x_10 writes a letter, then x_10 is poor, and that for all x_10, if x_10 were poor, then x_10 is writing a letter we can infer that for all x_10, x_10 will write a letter if and only if x_10 is poor via universal biconditional introduction. From the fact that for all x_10, if x_10 were an electrician, then x_10 is poor, and that for all x_10, if x_10 were poor, then x_10 is an electrician we can infer that for all x_10, x_10 is an electrician if and only if x_10 is poor via universal biconditional introduction. From the fact that for all x_10, x_10 will write a letter if and only if x_10 is poor, and that there is at least one x_10 for which x_10 will not write a letter we can infer that there is at least one x_10 for which x_10 is not poor via existential biconditional elimination. Finally, from the fact that for all x_10, x_10 is an electrician if and only if x_10 is poor, and that there is at least one x_10 for which x_10 is not poor we can infer that there is at least one x_10 for which x_10 is not an electrician via existential biconditional elimination. Therefore, the answer is yes.