The authors discuss complete interrelationships between eight main axioms of fuzzy implications. In particular, they discuss the following properties: neutrality of truth, exchange principle, ordering principle, strong negation principle, consequent boundary, identity principle, contrapositive symmetry and continuity, which are important in the characterizations of different classes of this family of fuzzy logic connectives. Each dependent case is presented with a proof or a citation, and each independent case is presented with a counterexample. The final results are also summarized in two tables.