Background: I am currently developing a general purpose programming language which allows formal verification (i.e. correctness proofs) of programs. During the development it came out that a lot of mathematics (order theory, lattice theory, cpos, etc.) is necessary to reach a sound definition.

For objects it is necessary to distinguish between identity and equality. Two objects are identical if they are indistinguishable. For this notion I have used the symbol $\sim$ (i.e. $a\sim b$ means the object $a$ is indistinguishable from the object $b$. Furthermore I allow user specific definitions of equality using the $=$ sign. I.e. $a=b$ means that $a$ and $b$ are equivalent with respect to some equivalence relation.

During my study of the needed mathematical theories I get more and more convinced that it would be better to take $=$ for identity or indistinguishability and $\sim$ for equalitiy or equivalence.

Is there a commonly accepted usage of symbols in mathematics expressing the notion of identy and equality?

P.S. For those who are interested in the definition of the programming language here is a link to my blog