I reading Lemma 17.16 of Chapter 17 of Gilbarg-Trudinger's elliptic PDEs book. The lemma is as follows
Let $u\in C^2(\Omega)$ satisfy $F[u]=0$ in $\Omega$ where $F$ is elliptic with respect to $u$. Then if $F\in C^{k}(\Gamma)$, $k\ge 1$, we have $u\in W^{k+2,p}_{loc}(\Omega)$ for all $p<\infty$; if $F\in C^k(\Gamma)$, $0<\alpha<1$, we have $u\in C^{k+2,\alpha}(\Omega)$.
For the proof, we consider the difference quotient $w(x)=\frac1h(u(x+he_l)-u(x))$ and see that $w$ satisfies $$a^{ij}D_{ij}w+b^iD_iw+cw=-f\quad(1)$$ with $a^{ij}=\int_0^1F_{ij}(x+\theta h,u_\theta,Du_\theta,D^2u_\theta)d\theta$, $b^i=\int_0^1F_{p_i}(x+\theta h,u_\theta,Du_\theta,D^2u_\theta)d\theta$, $c=\int_0^1F_{z}(x+\theta h,u_\theta,Du_\theta,D^2u_\theta)d\theta$ and $f=\int_0^1F_{x_l}(x+\theta h,u_\theta,Du_\theta,D^2u_\theta)d\theta$. It then applies $L^p$ estimates to $(1)$ and see that $\lvert D^2w\rvert_{p,\Omega^{\prime\prime}} $ is bounded in $L^p(\Omega^{\prime\prime})$ independent of $h$. Hence $u\in W^{3,p}_{loc}(\Omega)$ for any $p$ and $u\in C^{2,\alpha}(\Omega)$ by Sobolev embedding. The general case follows from bootstrapping.
I must have missed something but I was wondering isn't the $L^p$ estimates implying that $\lvert D^3u\rvert_{p,\Omega^{\prime\prime}} \le C$ where $C$ depends on $n,\lambda,\Lambda$ $\Omega^{\prime\prime}$, $\lvert F\rvert _{C^1}$ and $\lvert u\rvert _{C^2}$ and hence by Sobolev embedding $\lvert u\rvert _{C^{2,\alpha}}\le C $ which is the Evans-Krylov estimates Theorem 17.14?