Consider the following premises. There is at least one x_4 for which x_4 will climb a mountain. For all x_4, if x_4 plays squash, then x_4 is driving a car. For all x_4, x_4 will not climb a mountain or x_4 will play squash. For all x_4, if x_4 cooks, then x_4 is driving a car. Can we infer the following from them? If we can, name the inference rule being used: For all x_4, x_4 will not drive a car.
 From the fact that there is at least one x_4 for which x_4 will climb a mountain we can infer that there is at least one x_4 for which x_4 will climb a mountain or x_4 will cook via existential addition. From the fact that there is at least one x_4 for which x_4 will climb a mountain or x_4 will cook, and that for all x_4, x_4 will not climb a mountain or x_4 will play squash we can infer that there is at least one x_4 for which x_4 will cook or x_4 will play squash via existential resolution. Finally, from the fact that for all x_4, if x_4 cooks, then x_4 is driving a car, that for all x_4, if x_4 plays squash, then x_4 is driving a car, and that there is at least one x_4 for which x_4 will cook or x_4 will play squash we can infer that there is at least one x_4 for which x_4 will drive a car via existential disjunction elimination, which contradicts that For all x_4, x_4 will not drive a car. Therefore, the answer is no.