While studying the dynamics of the Einstein vacuum equations $$ \text{Ric}(g)=0 $$ for $(M,g)$ unknwon, I've come across the statement that in harmonic coordinates $x^\lambda$ defined by $\Box_g x^\lambda=0$ the principal part (all terms containing second derivaties of $g_{\mu\nu}$) of $\text{Ric}_{\mu\nu}$ is $$ g^{\alpha\beta}\partial_\alpha\partial_\beta g_{\mu\nu} $$ so that $\text{Ric}_{\mu\nu}=0$ becomes $$ g^{\alpha\beta}\partial_\alpha\partial_\beta g_{\mu\nu} = F(g,\partial g) $$ which is then quasi-linear.
Now I know that $$ \text{Ric}_{\mu\nu} = \partial_\alpha\Gamma^\alpha_{\mu\nu}-\partial_\nu\Gamma^\alpha_{\mu\alpha}+\Gamma^\alpha_{\alpha\beta}\Gamma^\beta_{\mu\nu}-\Gamma^\alpha_{\mu\beta}\Gamma^\beta_{\nu\alpha} $$ and also that in harmonic coordinates $g^{\alpha\beta}\Gamma^\lambda_{\alpha\beta}=0$ which can be seen when writing $\Box_g x^\lambda=0$ in local coordinates.
My question is: How does one get from this that the principal part of the Ricci is $g^{\alpha\beta}\partial_\alpha\partial_\beta g_{\mu\nu}$?