# Would this salvage the $\in|=$ exchange naive set theory?

This is a possible salvage for the failed attempt in this posting.

The salvage here is to require that every subformula $$\psi(y)$$ of $$\phi$$ having no parameter other than those in $$\phi$$, must satsify the antecdent of comprehension. To write this formally, it is:

Comprehension: If $$\phi$$ is a formula in the first order language of set theory (i.e.;$$\sf FOL(=,∈))$$, in which the symbol $$“x”$$ doesn’t occur free, and if $$\psi_1(y),..,\psi_n(y)$$ are all subformulas of $$\phi$$ in which $$y$$ is free, and having no parameter that is not a parameter of $$\phi$$; then: $$[\bigwedge_{i=1}^n \big{(}\exists y ((\psi_i(y))^=) \wedge \exists y ((\neg \psi_i(y))^=) \big{)} \to \exists x \forall y (y \in x \iff \phi)]$$ ; is an axiom.

Axiom of Multiplicity: $$\forall x,y \ \exists z (z \neq x \land z \neq y)$$

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I personally think this is complex a little bit, I highly doubt its consistency though. Yet if there is a chance that this is consistent, then it would actually prove all axioms of $$\sf NF$$, since full Extensionality is assumed here.