Here is how Darwall reconstructs Nagel’s argument.
Nagel starts by distinguishing the personal from the impersonal standpoints. Normative judgments from the personal standpoint are those of the form (1) I have reason to do x in C. Normative judgments from the impersonal standpoint are of the form (2) A has reason to do x in C. (Or, in the earlier lingo, (1a) my happiness is the sole good for me is a personal judgment; (1b) A’s happiness is the sole good for A is an impersonal judgment.)
Nagel believes that judgments (1) and (1a), made from the personal standpoint, are respectively equivalent to judgments (2) and (2a), made from the impersonal standpoint. Thus, A’s judgment I have reason to do x in C is no different from B’s judgment A has reason to do x in C. Likewise A’s judgment my happiness is the sole good for me has the same content as B’s judgment A’s happiness is the sole good for A.
Nagel’s view that correlate personal and impersonal judgments have the same content is unexceptionable. In effect, the only difference between the two is that each applies different names to the same entities. But the judgments are about the entities, not the names, so the truth does not vary with the changes in appellation. Or, as Nagel puts it, “[s]hifts of grammatical person … cannot be permitted to alter the sense of what is asserted about the circumstance which is the subject of the statement.”
Darwall summarizes Nagel’s thesis about the personal and impersonal standpoints thusly:
(A’) Judgments from the personal and impersonal standpoints must have the same content. That is, what is judged to be true of a practical situation is not affected merely by the standpoint from which it is judged.
The reason the truth of (A’) is supposed to spell trouble for egoism is that, conjoined with a plausible principle (which, on its face, the egoist qua egoist has no reason to deny), it implies that egoism is false. As Darwall puts it, the principle is that “practical judgments made from the personal standpoint have motivational content.” What this means is that if A judges that I have reason to do x in C, or my happiness is the sole good for me, that justifies A’s desiring to do x, or bring about only his own happiness. Or, put Darwall-wise,
(B’) The personal practical judgment (‘I have reason to do [x]’) has motivational content, i.e. a person’s acceptance of that judgment is sufficient to … justify his desiring to bring about what it recommends.
But if the personal practical judgment has motivational content, then, by (A’), so too should the impersonal practical judgment to which it is equivalent. In other words, if A’s personal judgment that “I have reason to do x in C” is sufficient to justify A’s desiring to do x in C, then, by (A’), B’s impersonal judgment that “A has reason to do x in C” is sufficient to justify B’s desiring to do x in C.
That this is so can be seen as follows. (A’) implies that the personal judgment and the impersonal judgment have equivalent content. (B’) entails that personal practical judgments (about the good, or what one ought to do) have motivational content, meaning that whoever accepts a personal practical judgment is justified in desiring to bring about what it recommends. But that means that impersonal judgments also have motivational content in the sense that whoever accepts them is justified in desiring to bring about what they recommend. So if B accepts the impersonal judgment that A has reason to do x in C, B is justified in altruistically desiring (what it recommends) that A does x in C.
In short, (A’) and (B’) jointly entail:
(C’) The impersonal practical judgment … must have motivational content, i.e. a person’s acceptance of that judgment is sufficient to … justify his desiring to bring about what it recommends.
(C’) is just the view that recognition of what is in others’ interests – of what is good for others – justifies wanting to help them. If another’s good is ipso facto your own then egoism is false.