Consider the following premises. For all x_10, if x_10 were a scientist, then x_10 is fast. For all x_10, x_10 is not a scientist. For all x_10, x_10 is fast if and only if x_10 will cook. There is at least one x_10 for which if x_10 were fast, then x_10 is a scientist. Can we infer the following from them? If we can, name the inference rule being used: For all x_10, x_10 will cook.
 From the fact that for all x_10, if x_10 were a scientist, then x_10 is fast, and that there is at least one x_10 for which if x_10 were fast, then x_10 is a scientist we can infer that there is at least one x_10 for which x_10 is a scientist if and only if x_10 is fast via existential biconditional introduction. From the fact that there is at least one x_10 for which x_10 is a scientist if and only if x_10 is fast, and that for all x_10, x_10 is not a scientist we can infer that there is at least one x_10 for which x_10 is not fast via existential biconditional elimination. Finally, from the fact that for all x_10, x_10 is fast if and only if x_10 will cook, and that there is at least one x_10 for which x_10 is not fast we can infer that there is at least one x_10 for which x_10 will not cook via existential biconditional elimination, which contradicts that For all x_10, x_10 will cook. Therefore, the answer is no.