This question is a follow up to this one, and relates to the answer to it. In some sense, this can be said to be the set-theoretic version of this question:
Let $\phi(x)$ be a formula in the language of set theory with one free variable $x$, let $\psi(x_1,\dots,x_n,z)$ be a formula with all free variables shown, and with $n>0$.
Let $T$ be theory consisting of the following single axiom: $$\forall x_1\dots\forall x_n \,[\phi(x_1) \land \dots \land \phi(x_n) \rightarrow \exists y\, \forall z \, (z \in y \leftrightarrow \psi(x_1,\dots,x_n,z))] $$
Now assume the following two conditions. First: $$T \vdash \forall x_1\dots,\forall x_n \forall y\, [\phi(x_1) \land \dots \land \phi(x_n) \land \forall z \, (z \in y \leftrightarrow \psi(x_1,\dots,x_n,z)) \rightarrow \phi(y)]$$ Second, the following sentence is not a logical validity: $$(\exists x\,\phi(x)) \rightarrow (\exists y\, \neg \phi(y))$$
Assuming these two conditions, is it always consistent to add the following sentence to $T$? $$ (\exists x\, \phi(x)) \rightarrow (\forall x\, \phi(x))$$