Consider the following premises. For all x_1, if x_1 were a carpenter, then x_1 is fast. There is at least one x_1 for which x_1 is poor. For all x_1, if x_1 were poor, then x_1 is a carpenter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_1 for which x_1 is fast.
 From the fact that for all x_1, if x_1 were poor, then x_1 is a carpenter, and that for all x_1, if x_1 were a carpenter, then x_1 is fast we can infer that for all x_1, if x_1 were poor, then x_1 is fast via universal transitivity. Finally, from the fact that for all x_1, if x_1 were poor, then x_1 is fast, and that there is at least one x_1 for which x_1 is poor we can infer that there is at least one x_1 for which x_1 is fast via existential modus ponens. Therefore, the answer is yes.