Axiom schema of replacement
Applications

The axiom schema of replacement is not necessary for the proofs of most theorems of ordinary mathematics. Indeed, Zermelo set theory (Z) already can interpret second-order arithmetic and much of type theory in finite types, which in turn are sufficient to formalize the bulk of mathematics. Although the axiom schema of replacement is a standard axiom in set theory today, it is often omitted from systems of type theory and foundation systems in topos theory.

At any rate, the axiom schema drastically increases the strength of ZF, both in terms of the theorems it can prove - for example the sets shown to exist - and also in terms of its proof-theoretic consistency strength, compared to Z. Some important examples follow:
oulipo:avoid_letter[t], combination:repeat_prompt