Error term in Bombieri-Vinogradov theorem

Question

In Primes in tuples I, in the proof of Proposition 5 (on p. 854) the authors use the "trivial estimate" $$ E'(N, q) \le \frac{2N}q \log N \quad \text{for $q \le N$}$$ where $$ E'(N, q) = \max_{(a,q)=1} \left| \vartheta(x;q,a) - \frac x{\varphi(q)} \right| $$ and $$ \vartheta(x;q,a) = \sum_{p \le x, \; p \equiv a \,\text{mod}\, q} \log p $$ as usual. I would like to understand how one obtains this estimate. I tried using the fact that $$ \liminf_{n \to \infty} \frac{\varphi(n) \log \log n}n = \mathrm e^{-\gamma} > 0. $$ However, I am hesitant to call this fact trivial and even if I do use it, in the original statement I get a constant which is worse than $2$. (I know that the constants don't really matter here, but it makes me feel like there should be a different argument.)