Consider the following premises. For all x_7, if x_7 cooks, then x_7 is reading a book. For all x_7, x_7 will drive a car or x_7 will cook. There is at least one x_7 for which if x_7 drives a car, then x_7 is reading a book. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_7 for which x_7 will read a book or x_7 is a mechanic.
 From the fact that there is at least one x_7 for which if x_7 drives a car, then x_7 is reading a book, that for all x_7, if x_7 cooks, then x_7 is reading a book, and that for all x_7, x_7 will drive a car or x_7 will cook we can infer that there is at least one x_7 for which x_7 will read a book via existential disjunction elimination. Finally, from the fact that there is at least one x_7 for which x_7 will read a book we can infer that there is at least one x_7 for which x_7 will read a book or x_7 is a mechanic via existential addition. Therefore, the answer is yes.