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Description logics (DLs) are a suitable formalism for representing knowledge about domains in which objects are described not only by attributes but also by binary relations between objects. Fuzzy extensions of DLs can be used for such domains when data and knowledge about them are vague and imprecise. One of the possible ways to specify classes of objects in such domains is to use concepts in fuzzy DLs. As DLs are variants of modal logics, indiscernibility in DLs is characterized by bisimilarity. The bisimilarity relation of an interpretation is the largest auto-bisimulation of that interpretation. In DLs and their fuzzy extensions, such equivalence relations can be used for concept learning. In this paper, we define and study fuzzy bisimulation and bisimilarity for fuzzy DLs under the Gödel semantics, as well as crisp bisimulation and strong bisimilarity for such logics extended with involutive negation. The considered logics are fuzzy extensions of the DL $\mathcal{ALC}_{reg}$ (a variant of PDL) with additional features among inverse roles, nominals, (qualified or unqualified) number restrictions, the universal role, local reflexivity of a role and involutive negation. We formulate and prove results on invariance of concepts under fuzzy (resp. crisp) bisimulation, conditional invariance of fuzzy TBoxex/ABoxes under bisimilarity (resp. strong bisimilarity), and the Hennessy-Milner property of fuzzy (resp. crisp) bisimulation for fuzzy DLs without (resp. with) involutive negation under the Gödel semantics. Apart from these fundamental results, we also provide results on using fuzzy bisimulation to separate the expressive powers of fuzzy DLs, as well as results on using strong bisimilarity to minimize fuzzy interpretations.