 The isomorphism  theorem (4)  above is a generalization of the isomorphism theorems we encountered earlier for algebraizable logics. What is interesting here is that the theorem holds for every logic system. Using  (2)  above,  theorem (4)  entails the isomorphism theorem for finitary and finitely algebraizable logics. Thus  theorem (4)  can be seen as the most general formulation of the mathematical logical phenomena that underlies the isomorphism theorems between the congruences of the algebras in a certain class and some kind of subsets of them we mentioned in  Section 9.