Has it been proved or disproved that for any fixed $a\geq 1$ there are infinitelly many primes $p_n\equiv a\pmod{n}$?

I believe i have proved that for every $a\geq1$ there are infinitelly many natural numbers $m$ with $m\equiv a\pmod{\pi(m)}$ where $\pi(m)$ is the prime counting function and i was wondering if this can be also true for a smallest subset of $\mathbb{N}$ (the primes).

OEIS does not tell much.http://oeis.org/A004648

Any reference would be appreciated.

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