The proof-theoretic strength of the axiom of extensionality, as well as those of other set-theoretic axioms (e.g., the axiom of regularity, foundation axioms, the axiom of choice, separation axioms), will be investigated, on quite weak settings with and without the axiom of infinity. The tool is mutual interpretability with subsystems of second order arithmetic (in the infinite case) or with two-sorted bounded arithmetic (in the non-infinite case). As a result, it will turn out that the axiom of extensionality is relatively stronger than other (weak) axioms and than previously considered.