Euler's totient formulation of the ternary Goldbach theorem with a congruence

Let $\phi$ denote Euler's totient function, and let $n$ be an odd integer greater than five. Consider integers $x,y$ that are non-negative. The system below is supplemented by a congruence modulo $3$:

$$\begin{cases} \phi(n-x-y)+1=n-x-y \\ \phi(2x-n)+1=2x-n \\ \phi(n-x+y)+1=n-x+y \\ (n-x-y)(2x-n) \equiv 0 \pmod{3} \end{cases}$$

Euler's totient formulation of the ternary Goldbach theorem with a congruence. The system has a solution $(x,y)$ in non-negative integers for every odd integer $n$ greater than five. The preceding theorem in the supplied text gives the same three totient equations without the congruence and identifies it with the ternary Goldbach theorem, which is stated there as a theorem and attributed to Helfgott. The text does not state a separate resolution of this strengthened formulation.