Universal-existential sentence ($\forall \mathbf{x}\exists \mathbf{y}\theta (\mathbf{x},\mathbf{y})$)

A universal-existential sentence has a universal block followed by an existential block and a quantifier-free formula. Such axioms are preserved under unions of chains of structures: the universally chosen tuple occurs in one stage, whose existential witnesses remain in the union. An inductive first-order theory has an axiomatization by these sentences.