Siegel-Walfisz theorem in proof of Vinogradov's three primes theorem

Question

Vinogradov's theorem states that for all sufficiently large odd numbers $N$ the equation $$ N = p_1 + p_2 + p_3 $$ has a solution in prime numbers. I was going through the proof of the theorem in this paper by Nicholas Rouse. At one point, while computing the integral over major arcs, I found this:
Lemma Let $\chi$ be a nontrivial Dirichlet character to modulus $q$. If $q \leqslant log(x)^M$ for a positive constant M, then $$ |\psi(x,\chi)| = O\left( x \exp(-C_M \sqrt{\log x}) \right) $$ for a positive constant $C_M$ which is a function of solely $M$. [Where $$ \psi(x,\chi) = \sum_{n \leqslant x} \chi(n) \Lambda(n) .$$] This seems like a version of Siegel-Walfisz theorem modified with the Dirichlet character. I was wondering, where could I see the proof of this fact? (I believe it shouldn't be that hard, but I don't want to get too technical)