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.

The result is true if you actually collapse the Woodin cardinal, not just the cardinals smaller than it. This follows from the results in Itay's chapter in the 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.