Consider the following premises. There is at least one x_5 for which if x_5 were a lawyer, then x_5 is climbing a mountain. For all x_5, x_5 is a lawyer or x_5 is not a doctor. For all x_5, if x_5 climbs a mountain, then x_5 is a doctor. Can we infer the following from them? For all x_5, if is not true that x_5 is a doctor if and only if x_5 is a lawyer.
 From the fact that there is at least one x_5 for which if x_5 were a lawyer, then x_5 is climbing a mountain, and that for all x_5, if x_5 climbs a mountain, then x_5 is a doctor we can infer that there is at least one x_5 for which if x_5 were a lawyer, then x_5 is a doctor. From the fact that for all x_5, x_5 is a lawyer or x_5 is not a doctor we can infer that for all x_5, if x_5 were a doctor, then x_5 is a lawyer. Finally, from the fact that for all x_5, if x_5 were a doctor, then x_5 is a lawyer, and that there is at least one x_5 for which if x_5 were a lawyer, then x_5 is a doctor we can infer that there is at least one x_5 for which x_5 is a doctor if and only if x_5 is a lawyer, which contradicts that For all x_5, if is not true that x_5 is a doctor if and only if x_5 is a lawyer. Therefore, the answer is no.