Let us consider the following stronger version of the Axiom Schema of Replacement (let us call it the Axiom Schema of Replacement for Definable Relations):

Let $\varphi$ be any formula in the language of ZF whose free variables are among the symbols $x,y,A,w_1,\dots,w_n$.

Then: $\forall w_1\dots\forall w_n [(\forall x\;(x\in A\to\exists y\; \varphi))\to\exists B\;\forall x(x\in A\to \exists y\; (y\in B\wedge \varphi))]$.

It can be shown that this Axiom Schema of Replacement for Definable Relations holds in ZF but for its proof one should apply the Axiom of Foundation (and the von Neumann cumulative hierarchy).

Question. Does the Axiom Schema of Replacement for Definable Relations follow from the axioms ZF with removed Axiom of Foundation?

Let us consider the following stronger version of the Axiom Schema of Replacement (let us call it the Axiom Schema of Replacement for Definable Relations):

Let $\varphi$ be any formula in the language of ZF whose free variables are among the symbols $x,y,A,w_1,\dots,w_n$.

Then: $\forall w_1\dots\forall w_n \forall A \;[(\forall x\;(x\in A\to\exists y\; \varphi))\to\exists B\;\forall x(x\in A\to \exists y\; (y\in B\wedge \varphi))]$.

It can be shown that this Axiom Schema of Replacement for Definable Relations holds in ZF but for its proof one should apply the Axiom of Foundation (and the von Neumann cumulative hierarchy).

Question. Does the Axiom Schema of Replacement for Definable Relations follow from the axioms ZF with removed Axiom of Foundation?