In this talk we continue the research in EQ-logics that can be
taken as special kind of fuzzy logics whose specific feature is
that they are based on equality (equivalence) as the basic
connective, while implication is derived. We develop formal system
of non-commutative predicate first-order EQ-logic we show some of
its main properties including the completeness theorem. Note that
in the predicate EQ-logic we consider also fuzzy equality between
objects. Emphasize that the approach taking equality exists also in
classical logic but the resulting logic is equivalent with the
other kinds of classical logics. In fuzzy logic, however, the
situation is different: Implication based and equality based
approaches are no more equivalent. EQ-logic makes it possible to
consider strong conjunction to be non-commutative and, unlike
non-commutative MTL-logic, still to have one implication only.