In his book "Some non-linear problems in Riemannian geometry" T.
Aubin states the following result (Theorem 3.56):

Let $A(u)=F(x,u,\nabla u,\nabla^2u)$ be a non-linear second order
differential operator defined on an open subset $\Omega$ of
$\mathbb{R}^n$; $F$ is an infinitely smooth function. Let $\Theta$
be a bounded subset of $C^2(\Omega)$. Suppose $A$ is uniformly
elliptic on $\Omega$ uniformly in $u\in \Theta$. Then if $A(\Theta)$
is bounded in $C^{r,\beta}(\Omega)$, then $\Theta$ is bounded in
$C^{r+2,\beta}(K)$ for any compact subset $K\subset \Omega$ (here
$r\geq 1$ is an integer, $\beta\in (0,1)$).

**My question is: whether this result is true indeed, and what is the
right reference?**

Aubin refers to two papers by L. Nirenberg:

(1) Comm. Pure Appl. Math.,6 (1953),103-156;

(2) Ann. Math. Studies 33, Princeton(1954), 95-100.

Also Aubin refers to previous results by other people containing
some weaker statements. I do not have the 1954 paper (and it does
not contain the detailed proof in fact), but the 1953 paper deals only with
the case of plane $n=2$. I need the case $n>2$. In the 1953 paper
Nirenberg mentions that he has generalized his result from $n=2$ to
higher dimensions, but in a somewhat weaker form. If I understand
correctly, **in addition he needs a bound on the modulus of continuity
of second derivatives of functions from the set $\Theta$. Was this
assumption removed since than?**