Consider the following premises. For all x_3, if x_3 cooks, then x_3 is a politician. For all x_3, if x_3 were an office clerk, then x_3 is fast. There is at least one x_3 for which x_3 is a politician if and only if x_3 is an office clerk. 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 if x_3 cooks, then x_3 is fast.
 From the fact that there is at least one x_3 for which x_3 is a politician if and only if x_3 is an office clerk we can infer that there is at least one x_3 for which if x_3 were a politician, then x_3 is an office clerk via existential biconditional elimination. From the fact that there is at least one x_3 for which if x_3 were a politician, then x_3 is an office clerk, and that for all x_3, if x_3 were an office clerk, then x_3 is fast we can infer that there is at least one x_3 for which if x_3 were a politician, then x_3 is fast via existential transitivity. Finally, from the fact that for all x_3, if x_3 cooks, then x_3 is a politician, and that there is at least one x_3 for which if x_3 were a politician, then x_3 is fast we can infer that there is at least one x_3 for which if x_3 cooks, then x_3 is fast via existential transitivity. Therefore, the answer is yes.