Schauder estimate for $f \in L^\infty$

I was reading an article where at some point the author uses the following estimate:

Let $u$ be a solution of$$\Delta u = f \quad \text{in } B_1$$for $f \in L^\infty$. Then $u \in C^{1,1 - \varepsilon}$ inside $B_1$ for any $\varepsilon \in (0,1)$ and satisfies$$\|u\|_{C^{1,1 - \varepsilon}(B_{1/2})} \le C_\varepsilon \left(\|f\|_{L^\infty(B_{1})} + \|u\|_{L^\infty(B_{1})}\right).$$

It looks like a variation of Schauder estimate for $f \in L^\infty$ instead of $f \in C^{0, \alpha}$. I was trying to find a proof of this result, but I didn't find any complete argument, only sketches where I struggled to fill the holes. Does anyone have a reference where the author proves this result?