This question is more of a trick.
Notation: for any formula $\phi$, let $\phi^=$ mean the formula obtained by merely replacing each occurrence of the membership symbol $\in$ in $\phi$ by the equality symbol $=$, so for example if $\phi$ is the formula $(y \in y)$, then $\phi^=$ is the formula $(y=y)$.
We assume full Extensionality throughout this exposition.
Notice this naive like Comprehension axiom scheme:
If $\phi$ is a formula in the first order language of set theory (i.e.; $\sf FOL(=,\in)$), in which the symbol $``x"$ doesn't occur free, then: $$[\exists y (\phi^=) \wedge \exists y ((\neg \phi)^=) \to \exists x \forall y (y \in x \iff \phi)]$$; is an axiom.
Axiom of Multiplicity: $\exists x,y: x \neq y $
This theory supply the appearance of an inconsistent theory. However, a proof of this inconsistency keeps eluding me?!
Is this theory consistent?
The point is that this theory is very naive. The antecedent of comprehension is an extremely simple syntactical procedure, more of a trick really! In other words its a very naive kind of restriction on the original unrestricted naive comprehension.
The general expectation for such maneuvers is inconsistency!?
This theory [if consistent] has a universal set. So if consistent, it might be interpretable in one of the fragments of $\sf NF$? It would be nice to see if that is the case!