Just as prudence aims at one agent's happiness, an impartial moral point of view aims at happiness without restriction to any particular agent.

On this reading, Mill is not trying to derive utilitarianism from egoism (see Hall 1949). Rather, he is assuming that the moral point of view is impartial in a way that prudence is not. Just as prudence aims at the agent’s own happiness, so too, Mill thinks, morality, which is impartial, aims at happiness as such. On this reading, the structure of Mill’s proof looks something like this.