Every expository article on hyperkähler manifolds that I have read states without detailed proof the following fact:

**It follows from Yau's theorem (i.e. a compact Kähler manifold $M$ with $c_1(M)=0$ admits a Ricci-flat Kähler metric) that if a compact Kähler manifold $M$ has a complex symplectic form $\omega_\mathbb{C}$ then there exists a Kähler form $\omega$ on $M$ such that $\omega$ and $\omega_\mathbb{C}$ constitute a hyperkähler structrue.**

I have trouble trying to see this. Indeed, for the metric to be hyperkähler we require the holonomy group to be in $Sp(n)$ (where $\mathrm{dim}_\mathbb{R}M=4n$), so at least the holonomy should be in $Sp(2n,\mathbb{C})$, which means that the Levi-Civita covariant derivative of $\omega_\mathbb{C}$ is $0$. This is a condition stronger than Ricci-flatness at first glance and I can't see how to realize this only knowing the existence of Ricci-flat metric.