Is my process of doing this math accurate?
Solution 1:
The symbol $≡$ has multiple meanings in mathematics, to indicate a definition, logical equivalence between two sentences, an equivalence relation (like in modular arithmetic, and congruence of geometric figures), an identity (an equation like $x^2-y^2≡(x+y)(x-y)$ that is true for every allowable combination of variable values), etc.
In your quoted example, there is no compelling reason to switch $=$ and $≡.$ The latter is just confusing, unless the author has prefaced it with a note on their particular usage of $≡.$