Numerical identity entails absolute indiscernibility. If x and y are indiscernible, they share all the same traits and x = y. Identical twins are not one and the same and thus are not numerically identical; they are instead extremely similar, thus sharing qualitative identity.
For every x, for every y,
If x = y, then x and y have exactly the same properties.
Leibniz’s Law of Indiscernibility of the Identical is useful in demonstrating the distinctness of two easily confused things. For example, some materialists argue that mental phenomena are simply neurophysiological phenomena. Their opponents typically attempt to show some mental phenomena have properties that do not occur neurophysiologically. If their opponents succeeded, then Liebniz’s Law of Indiscernibility of the Identical woud entail, via Universal Instantiation and Modus Tollens, that mental activities are not identical with neurophysiological activities.
For every x, for every y,
If x and y have exactly the same properties, then x = y.
This argument does not prove that identical twins are identical. To be numerically identical, identical twins would have to share the same spatial location, a feat which is possible for existing objects.