If a == b iff a === b, then == ~ ===. If == and === are different (which they presumable are, because otherwise we would only use one symbol) then by definition there exist counter-examples of the thing you were trying to _always_ satisfy.

In other words, it would never even occur to a mathematician to make your mistake...

More to the point, a mathematician wouldn't assume that some arbitrary predicate that you are allowed to define at will behaves the same way as a system-defined equivalence operator.

Edit: Added last paragraph and readability changes. Also, ~ is another symbol for equality, this time equality over binary relations.