I have a question related to the Bombieri-Vinogradov Theorem and the Goldbach conjecture. Bombieri-Vinogradov theorem states that if $ A > 0 $ then for all $Q$ such that $ x^{1/2} (\log x )^{-A} \leq Q \leq x^{1/2} $ we have that $$ \sum_{ q \leq Q} \max_{ (a,q)=1} \max_{y \leq x} \left| \phi(y;q,a) - y/\varphi(q) \right| \ll x^{1/2} Q (\log x)^5 $$
In the class professor did a remark: A. Renyi proved a first result of weak version of Goldbach conjecture that every sufficiently large even integer can be written as $p + p_1 p_2 \ldots p_r $ for some bounded constant $r$. The Bombieri-Vinogradov Theorem would allow $r=3$ and the best current result is $r=2$ due to Chen.
My question: I wonder why Bombieri-Vinogradov Theorem allows $r=3$, I don't see why.