Full Idea
The derivability of Peano's Postulates from Hume's Principle in second-order logic has been dubbed 'Frege's Theorem', (though Frege would not have been interested, because he didn't think Hume's Principle gave an adequate definition of numebrs).
Gist of Idea
Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle
Source
A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8 n1)
Book Reference
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.215
A Reaction
Frege said the numbers were the sets which were the extensions of the sets created by Hume's Principle.