Consider the following premises. For all x_9, if x_9 writes a letter, then x_9 is a lawyer. There is at least one x_9 for which x_9 will write a letter or x_9 is sad. For all x_9, if x_9 were a nurse, then x_9 is a lawyer. For all x_9, if x_9 were sad, then x_9 is a nurse. For all x_9, if x_9 were a lawyer, then x_9 is writing a letter. Can we infer the following from them? If we can, name the inference rule being used: For all x_9, x_9 is not a lawyer.
 From the fact that for all x_9, if x_9 writes a letter, then x_9 is a lawyer, and that for all x_9, if x_9 were a lawyer, then x_9 is writing a letter we can infer that for all x_9, x_9 will write a letter if and only if x_9 is a lawyer via universal biconditional introduction. From the fact that for all x_9, x_9 will write a letter if and only if x_9 is a lawyer we can infer that for all x_9, if x_9 writes a letter, then x_9 is a lawyer via universal biconditional elimination. From the fact that for all x_9, if x_9 were sad, then x_9 is a nurse, and that for all x_9, if x_9 were a nurse, then x_9 is a lawyer we can infer that for all x_9, if x_9 were sad, then x_9 is a lawyer via universal transitivity. Finally, from the fact that for all x_9, if x_9 writes a letter, then x_9 is a lawyer, that for all x_9, if x_9 were sad, then x_9 is a lawyer, and that there is at least one x_9 for which x_9 will write a letter or x_9 is sad we can infer that there is at least one x_9 for which x_9 is a lawyer via existential disjunction elimination, which contradicts that For all x_9, x_9 is not a lawyer. Therefore, the answer is no.