You asked for a "law" of equality (whatever that is) and provided an answer that proves it converges to 1. What more could you possibly want?
For ordinary math, though, using some criterion for equality (for example x>=y and y>=x) is basic and not controversial. So it seems unconvincing (to me) when you seem to imply the opposite.
I use applied math. I haven't taken a class in real analysis. But it's fun how often grinding out the solution to a "real world," practical PDE turns out not to actually be the nicest (simplest and/or clearest and/or sufficiently insight-producing) way to understand the (hopefully) corresponding physical problem in the lab.
Stripping off the "calculus" and replacing it by limits sometimes seems to help highlight alternate perspectives that the magic "integrals" and "derivatives" kind of conceal.
Even when it's not more effective, it's definitely more fun.
You are putting words in my mouth.
And you clearly do not understand the answer.
I guess I'm not very good at ELI5 because I very clearly answered your question with your own proposal.
Maybe when you get to college a professor can do a better job explaining it to you (if you actually make it to college, because you're going to struggle very hard if that's how you think when an answer is spoon-fed to you).