Consider the following premises. For all x_7, if x_7 were fast, then x_7 is a doctor. For all x_7, x_7 will read a book if and only if x_7 is a doctor. For all x_7, x_7 will read a book or x_7 will cook. For all x_7, if x_7 cooks, then x_7 is fast. Can we infer the following from them? If we can, name the inference rule being used: For all x_7, x_7 is a doctor.
 From the fact that for all x_7, x_7 will read a book if and only if x_7 is a doctor we can infer that for all x_7, if x_7 reads a book, then x_7 is a doctor via universal biconditional elimination. From the fact that for all x_7, if x_7 cooks, then x_7 is fast, and that for all x_7, if x_7 were fast, then x_7 is a doctor we can infer that for all x_7, if x_7 cooks, then x_7 is a doctor via universal transitivity. Finally, from the fact that for all x_7, if x_7 reads a book, then x_7 is a doctor, that for all x_7, if x_7 cooks, then x_7 is a doctor, and that for all x_7, x_7 will read a book or x_7 will cook we can infer that for all x_7, x_7 is a doctor via universal disjunction elimination. Therefore, the answer is yes.