Consider the following premises. There is at least one x_10 for which x_10 is sad or x_10 will cook. For all x_10, if x_10 cooks, then x_10 is an astronaut. For all x_10, if x_10 were a doctor, then x_10 is an astronaut. For all x_10, x_10 will play squash. For all x_10, if x_10 were sad, then x_10 is a doctor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which x_10 will play squash and x_10 is an astronaut.
 From the fact that for all x_10, if x_10 were sad, then x_10 is a doctor, and that for all x_10, if x_10 were a doctor, then x_10 is an astronaut we can infer that for all x_10, if x_10 were sad, then x_10 is an astronaut via universal transitivity. From the fact that for all x_10, if x_10 were sad, then x_10 is an astronaut, that for all x_10, if x_10 cooks, then x_10 is an astronaut, and that there is at least one x_10 for which x_10 is sad or x_10 will cook we can infer that there is at least one x_10 for which x_10 is an astronaut via existential disjunction elimination. Finally, from the fact that for all x_10, x_10 will play squash, and that there is at least one x_10 for which x_10 is an astronaut we can infer that there is at least one x_10 for which x_10 will play squash and x_10 is an astronaut via existential conjunction. Therefore, the answer is yes.