Consider the following premises. For all x_4, x_4 is a carpenter if and only if x_4 is a police officer. For all x_4, x_4 is not a carpenter. For all x_4, if x_4 were a police officer, then x_4 is a carpenter. Can we infer the following from them? There is at least one x_4 for which x_4 is a police officer.
 From the fact that for all x_4, x_4 is a carpenter if and only if x_4 is a police officer we can infer that for all x_4, if x_4 were a carpenter, then x_4 is a police officer. From the fact that for all x_4, if x_4 were a carpenter, then x_4 is a police officer, and that for all x_4, if x_4 were a police officer, then x_4 is a carpenter we can infer that for all x_4, x_4 is a carpenter if and only if x_4 is a police officer. Finally, from the fact that for all x_4, x_4 is a carpenter if and only if x_4 is a police officer, and that for all x_4, x_4 is not a carpenter we can infer that for all x_4, x_4 is not a police officer, which contradicts that There is at least one x_4 for which x_4 is a police officer. Therefore, the answer is no.