Consider the following premises. For all x_4, if x_4 were a doctor, then x_4 is driving a car. For all x_4, if x_4 drives a car, then x_4 is taking a plane. There is at least one x_4 for which x_4 is a doctor or x_4 will not take a plane. Can we infer the following from them? If we can, name the inference rule being used: For all x_4, if is not true that x_4 will take a plane if and only if x_4 is a doctor.
 From the fact that there is at least one x_4 for which x_4 is a doctor or x_4 will not take a plane we can infer that there is at least one x_4 for which if x_4 takes a plane, then x_4 is a doctor via existential rewriting. From the fact that for all x_4, if x_4 were a doctor, then x_4 is driving a car, and that for all x_4, if x_4 drives a car, then x_4 is taking a plane we can infer that for all x_4, if x_4 were a doctor, then x_4 is taking a plane via universal transitivity. Finally, from the fact that there is at least one x_4 for which if x_4 takes a plane, then x_4 is a doctor, and that for all x_4, if x_4 were a doctor, then x_4 is taking a plane we can infer that there is at least one x_4 for which x_4 will take a plane if and only if x_4 is a doctor via existential biconditional introduction, which contradicts that For all x_4, if is not true that x_4 will take a plane if and only if x_4 is a doctor. Therefore, the answer is no.