Consider the following premises. For all x, if x were an astronaut, then x is writing a letter. There is at least one x for which x is a nurse or x will not write a letter. For all x, if x were a nurse, then x is an astronaut. For all x, x will not write a letter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which x is not a nurse.
 From the fact that there is at least one x for which x is a nurse or x will not write a letter we can infer that there is at least one x for which if x writes a letter, then x is a nurse via existential rewriting. From the fact that for all x, if x were a nurse, then x is an astronaut, and that for all x, if x were an astronaut, then x is writing a letter we can infer that for all x, if x were a nurse, then x is writing a letter via universal transitivity. From the fact that there is at least one x for which if x writes a letter, then x is a nurse, and that for all x, if x were a nurse, then x is writing a letter we can infer that there is at least one x for which x will write a letter if and only if x is a nurse via existential biconditional introduction. Finally, from the fact that there is at least one x for which x will write a letter if and only if x is a nurse, and that for all x, x will not write a letter we can infer that there is at least one x for which x is not a nurse via existential biconditional elimination. Therefore, the answer is yes.