Consider the following premises. There is at least one x_11 for which x_11 will work or x_11 will write a letter. For all x_11, if x_11 works, then x_11 is playing tennis. For all x_11, if x_11 writes a letter, then x_11 is playing tennis. For all x_11, x_11 is a carpenter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_11 for which x_11 is a carpenter.
 From the fact that for all x_11, if x_11 works, then x_11 is playing tennis, that for all x_11, if x_11 writes a letter, then x_11 is playing tennis, and that there is at least one x_11 for which x_11 will work or x_11 will write a letter we can infer that there is at least one x_11 for which x_11 will play tennis via existential disjunction elimination. From the fact that there is at least one x_11 for which x_11 will play tennis, and that for all x_11, x_11 is a carpenter we can infer that there is at least one x_11 for which x_11 will play tennis and x_11 is a carpenter via existential conjunction. Finally, from the fact that there is at least one x_11 for which x_11 will play tennis and x_11 is a carpenter we can infer that there is at least one x_11 for which x_11 is a carpenter via existential simplification. Therefore, the answer is yes.