I appreciate if you could help me to find a reference (and a proof).

Combining Dirichlet theorem on primes in arithmetic progression with Chebotarev densitiy theorem, we know that given two positive integers $a$ and $m$ with $\gcd(a,m)=1$ the proportion of primes $p$ with $p\cong a\mod m$ is $1/\varphi(m)$, where $\varphi$ is the Euler totient function.

Can you give me a reference (and possibly proof) of a quantitative version of this result? That is, a result giving an EXPLICIT function $\Phi(m)$ so that

$$\frac{|\{p\mid p\textrm{ prime with }p\leq n \textrm{ and }p\equiv a\mod m\}|}{|\{p\mid p\textrm{ prime with }p\leq n \}|}\geq \frac{n}{\log(n)\Phi(m)}.$$