Consider the following premises. For all x_5, if x_5 drinks water, then x_5 is listening to a song. For all x_5, if x_5 were a soldier, then x_5 is a doctor. For all x_5, if x_5 listens to a song, then x_5 is a soldier. For all x_5, if x_5 were a doctor, then x_5 is drinking water. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which if is not true that x_5 is a doctor if and only if x_5 will drink water.
 From the fact that for all x_5, if x_5 drinks water, then x_5 is listening to a song, and that for all x_5, if x_5 listens to a song, then x_5 is a soldier we can infer that for all x_5, if x_5 drinks water, then x_5 is a soldier via universal transitivity. From the fact that for all x_5, if x_5 drinks water, then x_5 is a soldier, and that for all x_5, if x_5 were a soldier, then x_5 is a doctor we can infer that for all x_5, if x_5 drinks water, then x_5 is a doctor via universal transitivity. Finally, from the fact that for all x_5, if x_5 were a doctor, then x_5 is drinking water, and that for all x_5, if x_5 drinks water, then x_5 is a doctor we can infer that for all x_5, x_5 is a doctor if and only if x_5 will drink water via universal biconditional introduction, which contradicts that There is at least one x_5 for which if is not true that x_5 is a doctor if and only if x_5 will drink water. Therefore, the answer is no.