As part of a [proof](https://math.stackexchange.com/a/912046/160028) of the CRT, I've encountered the following:

We want to find a solution for the system

$$x \equiv a\_1 \pmod{n\_1} \\ x \equiv a\_2 \pmod{n\_2}$$

Then the author look for a solution of the form $x= a\_1 +dy$ and claims that the system is equivalent to:

$$dy \equiv 0 \pmod{n\_1} \\ dy \equiv a\_2 - a\_1 \pmod{n\_2}$$

In the second equation it seems like he was able to "move" the $a\_1$ term to the other side of the equivalence - Why is it valid?