This follows from the results in Itay's chapter in Handbook. See MR2768701, zbM1198.03057
Neeman, Itay. Determinacy in $L(\mathbb R)$. In "Handbook of set theory. Vols. 1, 2, 3", 1877–1950. Springer, Dordrecht, 2010 ISBN:978-1-4020-4843-2
Particularly, see Corollary 6.12 and, really, Chapter 6, which "localizes" the results of Chapter 5 (which, in turn, assume that there is a measurable cardinal above the Woodin, and show that in $V$ we have $\mathbf\Sigma^1_2$-determinacy). The result there is stated with $\Delta^1_2$-determinacy in the conclusion, but Martin proved that if $\mathsf{DC}$ holds, then $\Delta^1_2$-determinacy gives $\Sigma^1_2$-determinacy. (Note these are lightface results). In turn, Martin's theorem is Theorem 6.3 in the Handbook chapter by Peter and Hugh, see MR2768702, zbM1198.03072
Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In "Handbook of set theory. Vols. 1, 2, 3", 1951–2119, Springer, Dordrecht, 2010.