Consider the following premises. For all x_7, if x_7 works, then x_7 is cooking. There is at least one x_7 for which if x_7 cooks, then x_7 is driving a car. For all x_7, x_7 will sleep or x_7 will not drive a car. 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 if x_7 works, then x_7 is sleeping.
 From the fact that for all x_7, x_7 will sleep or x_7 will not drive a car we can infer that for all x_7, if x_7 drives a car, then x_7 is sleeping via universal rewriting. From the fact that for all x_7, if x_7 works, then x_7 is cooking, and that there is at least one x_7 for which if x_7 cooks, then x_7 is driving a car we can infer that there is at least one x_7 for which if x_7 works, then x_7 is driving a car via existential transitivity. Finally, from the fact that there is at least one x_7 for which if x_7 works, then x_7 is driving a car, and that for all x_7, if x_7 drives a car, then x_7 is sleeping we can infer that there is at least one x_7 for which if x_7 works, then x_7 is sleeping via existential transitivity. Therefore, the answer is yes.