In this paper we continue development of formal theory of a special class of fuzzy LOGICS, called EQ-LOGICS. Unlike fuzzy LOGICS being extensions of the MTL-logic in which the basic connective is implication, the basic connective in EQ-LOGICS is equivalence. Therefore, a new algebra of truth values called EQ-algebra was developed. This is a lower semi lattice with top element endowed with two binary operations of fuzzy equality and multiplication. EQ-algebra generalizes residuated lattices, namely, every residuated lattice is an EQ-algebra but not vice-versa.In this paper, we introduce additional connective D in EQ-LOGICS (analogous to Baaz delta connective in MTL-algebra based fuzzy LOGICS) and demonstrate that the resulting logic has again reasonable properties including completeness.Introducing D in EQ-logic makes it possible to prove also generalized deduction theorem which otherwise does not hold in EQ-LOGICS weaker than MTL-logic.