Consider the following premises. There is at least one x_3 for which x_3 is an astronaut or x_3 is not a scientist. For all x_3, if x_3 were a nurse, then x_3 is a scientist. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_3 for which if x_3 were a nurse, then x_3 is an astronaut.
 From the fact that there is at least one x_3 for which x_3 is an astronaut or x_3 is not a scientist we can infer that there is at least one x_3 for which if x_3 were a scientist, then x_3 is an astronaut via existential rewriting. Finally, from the fact that for all x_3, if x_3 were a nurse, then x_3 is a scientist, and that there is at least one x_3 for which if x_3 were a scientist, then x_3 is an astronaut we can infer that there is at least one x_3 for which if x_3 were a nurse, then x_3 is an astronaut via existential transitivity. Therefore, the answer is yes.