Consider the following premises. For all x_7, x_7 will play squash. For all x_7, x_7 will write a letter or x_7 is fast. There is at least one x_7 for which x_7 will not write a letter. Can we infer the following from them? If we can, name the inference rule being used: For all x_7, x_7 is not fast.
 From the fact that for all x_7, x_7 will write a letter or x_7 is fast, and that there is at least one x_7 for which x_7 will not write a letter we can infer that there is at least one x_7 for which x_7 is fast via existential disjunctive syllogism. From the fact that for all x_7, x_7 will play squash, and that there is at least one x_7 for which x_7 is fast we can infer that there is at least one x_7 for which x_7 will play squash and x_7 is fast via existential conjunction. Finally, from the fact that there is at least one x_7 for which x_7 will play squash and x_7 is fast we can infer that there is at least one x_7 for which x_7 will play squash via existential simplification, which contradicts that For all x_7, x_7 is not fast. Therefore, the answer is no.