I'm more familiar with the notation $a \equiv b \pmod c$, but I think this is equivalent to $a \bmod c = b \bmod c $, which makes it clear that we should put a $=$ instead of $\equiv$.
What's the reason for the change of sign? If it's to emphasize that modular equivalence is a congruence relation, why don't we use the $\equiv$ sign in both notations?
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