Is this generalization of CRT true?

Question

$a^n \equiv b_1 \bmod p_1$

$a^n \equiv b_2 \bmod p_2$

if $gcd(b_1,p_2) = gcd(b_2,p_1) = 1$ then $a^n \equiv b_1 \cdot b_2 \bmod p_1\cdot p_2$

$p_1$ and $p_2$ are not necessarily coprime.

edit

what about $gcd(p_1, p_2) = 1$ ?