On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom

Examples of effectively indiscernible projective sets real numbers in various models set theory are presented. We prove that it is true, Miller and Laver generic extensions the constructible universe, there exists a lightface ?21 equivalence relation on all nonconstructible reals, having exactly two classes, neither one which ordinal definable, therefore classes OD-indiscernible. A similar but somewhat weaker result obtained for Silver extensions. The other main any n, starting with 2, existence pair countable disjoint OD-indiscernible sets, whose associated belongs to ?n1, does not imply such ?n1 or lower class.