Equality symbols in modular arithmetic

E.g., can I write $(a^{p})^{2p} \equiv a^{2p}=a^pa^p\equiv aa\equiv a^2\pmod{\! p}$?

I often see equality symbols inbetween mod equivalences. The equality signs point out the equality is not restricted to mod $p$ and holds in general.

Can it technically be used this way? I know it is not ambiguous, it is understandable and may make some things clearer, and I know this question is nitpicking about notation. But I'm thinking maybe there is some consensus regarding this?