The following theory is a fragment of $\small \sf NF$. My question is about if it is known to be consistent without assuming the consistency of $\small\sf NF$.
The language is of first order logic with equality and membership.
1. Extnesionality: $\forall x \forall y (\forall z ( z \in x \leftrightarrow z \in y ) \to x=y)$
2. Comprehension: if $\phi(y)$ is a formula in which $y$ only occurs free, and $x$ not occurring free, having no more than one occurrence of an atomic sub-formula that contains a bound variable in $\phi(y)$ and $y$, and having no more than one occurrence of an atomic sub-formula containing two bound variables in $\phi(y)$, otherwise all atomic sub-formulas must contain a parameter; then all closures of: $$\exists x \forall y (y \in x \leftrightarrow \phi(y))$$; are axioms.
Terminology: by atomic formula here it is meant a formula having exactly two occurrences of variable symbols and one primitive predicate symbol, i.e. either $\in$ or $=$, in between. By parameters of a formula $\phi(y)$ its meant any free variable symbol in $\phi(y)$ other than the symbol $``y"$. The primitive logical connectives are the customary four ones and the bi-conditional.
/Theory definition finished.
Now this theory is a fragment of $\small \sf NF$, since all so qualified formulas in comprehension are weakly stratified! IF we in addition allow for the formula to have two atomic sub-formulas with bound variables (instead of one) provided that the total number of variables in those two atomic sub-formulas must be more than two, then we get full $\small \sf NF$.
My guess is that its very weak and thus consistent.