Consider the following premises. For all x, x will drive a car if and only if x is a nurse. There is at least one x for which x will drive a car if and only if x will write a letter. Can we infer the following from them? For all x, the fact that x writes a letter does not imply that x is a nurse.
 From the fact that there is at least one x for which x will drive a car if and only if x will write a letter we can infer that there is at least one x for which if x writes a letter, then x is driving a car. Finally, from the fact that for all x, x will drive a car if and only if x is a nurse we can infer that for all x, if x drives a car, then x is a nurse, which contradicts that For all x, the fact that x writes a letter does not imply that x is a nurse. Therefore, the answer is no.