Consider the following premises. For all x_10, x_10 is a musician. For all x_10, if x_10 were a musician, then x_10 is writing a letter. There is at least one x_10 for which if x_10 writes a letter, then x_10 is cooking. 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 cook or x_10 is an astronaut.
 From the fact that for all x_10, if x_10 were a musician, then x_10 is writing a letter, and that for all x_10, x_10 is a musician we can infer that for all x_10, x_10 will write a letter via universal modus ponens. From the fact that there is at least one x_10 for which if x_10 writes a letter, then x_10 is cooking, and that for all x_10, x_10 will write a letter we can infer that there is at least one x_10 for which x_10 will cook via existential modus ponens. Finally, from the fact that there is at least one x_10 for which x_10 will cook we can infer that there is at least one x_10 for which x_10 will cook or x_10 is an astronaut via existential addition. Therefore, the answer is yes.