This follows from the results in Itay's chapter in Handbook. See [MR2768701][1], [zbM1198.03057][2]

> 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][3], [zbM1198.03072][4]
> Koellner, Peter; Woodin, W. Hugh. *Large cardinals from determinacy*. In "Handbook of set theory. Vols. 1, 2, 3", 1951–2119, Springer, Dordrecht, 2010.


  [1]: https://mathscinet.ams.org/mathscinet/relay-station?mr=2768701
  [2]: https://zbmath.org/1198.03057
  [3]: https://mathscinet.ams.org/mathscinet/relay-station?mr=2768702
  [4]: https://zbmath.org/1198.03072