We show that the Axiom of Foundation, as well as the Antifoundation Axiom AFA, plays a crucial role in determining the decidability of the following problem. Given a first-order theory T over the language $ =,\in$, and a sentence F of the form $ \forall x_1, \ldots, x_n \exists y F^M$ with $ F^M$ quantifier-free in the same language, are there models of T in which F is true? Furthermore we show that the Extensionality Axiom is quite irrelevant in that respect.
"The Decidability of the $ \forall^*\exists$ Class and the Axiom of Foundation." Notre Dame J. Formal Logic 42 (1) 41 - 53, 2001. https://doi.org/10.1305/ndjfl/1054301354