In mathematical terms, what I'm saying is that equality is more fundamental than equivalence as defined by some other equivalence relation[1]. If you're considering integers according to their remainder when divided by 5, you can easily show that 4 is equal to 4, and that 4 is equal to -6. If you change your metric to remainder when divided by 3, you'll notice that 4 is still equal to 4, but no longer equal to -6. That is because 4 is fundamentally more similar to 4 than -6 is.

[1] And as I've already mentioned, you can see that this is true by looking at the definition of an equivalence relation. They all treat everything as equivalent to itself.