Uh....the problem is, it doesn't say that. You add to it by your own imagination.
Act 13:32 And we declare unto you glad tidings, how that the promise which was made unto the fathers,
Act 13:33 God hath fulfilled the same unto us their children, in that he hath raised up Jesus again; as it is also written in the second psalm, Thou art my Son, this day have I begotten thee.
There is nothing imagined. He goes on to explain in his excitement about his gospel that because of the resurrection, because the promises to Israel shifted to Christ (quoting isaiah), those who believe know that they are justified, and that is the promise to Israel.
That is why this is a 'short' version of Gal 3, saying the same thing.
It is you who do not know that Paul was preaching this everywhere as his Gospel.
As for the 'proof' theme (the resurrection proved that justification was secure), he said the same thing in Rom 4:25 where it is as clear as can be because of a parallel cause-and-effect about sin:
he was put to death because of our sin
he was raised because of our justification.
Paul was not in the business of saying that the resurrection was one of the things that needed to be done for it (justification of sinners) but that it proved it. That's why Ps 16 is quoted so much: God would not let his perfect one suffer decay. He was awarded resurrection and ascension and enthronement, which is what the apostles screamed all over Israel for months. He is enthroned in the David sense, and Paul says that is exactly according his Gospel in Rom 1's intro.
There he says that the resurrection DECLARED that he was the Son of God. RIght: the event itself made the declaration. That is the power of apostolic preaching. Romans ends with a parallel: the incoming (believing) of the nations DECLARES what the OT is about. It is not a doctrine in a passage as much as it is a well-known and publically noticed miracle. That is what was embedded in the OT ALL THIS TIME. And now (Paul's now) it is actually happening.
