**Replacement:** if $\phi(x,z)$ is a formula in which all and only symbols $``x,z,x\_1,..,x\_n"$ occur free, and non of them occur as bound, and in which the symbol $``B"$ never occur; then: $$\forall x\_1,..,x\_n \\\forall A \exists B \forall y \ [y \in B \leftrightarrow \exists x \in A \forall z (\phi(x,z) \leftrightarrow z=y)]$$; is an axiom.


The above scheme is a just a reformulation of the axiom schema of replacement.


**Cyclic replacement** is the above schema but with allowing the symbol $``B"$ to occur free in $\phi(x,z)$.



> 
> 1. Is it inconsistent to replace axiom schema of replacement in ZFC-Foundation by axiom schema of cyclic replacement?
> 2. Would that be inconsistent with Aczel's anti-foundation axiom?
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Is cyclic replacement inconsistent with ZFC-Foundation?
Unless I'm misunderstanding something here, Cyclic Replacement is inconsistent with just about any set theory. If $p$ and $q$ are two distinct sets, let $\phi(x,z,B)$ be $(z=p\land B\neq\{p\})\lor(z=q\land B=\{p\})$. Let $A$ be your favorite nonempty set. (Since you insist that $\phi$ contain $x$ free, add a clause "$\land(x=x)$".)Then cyclic replacement says there's a set $B$ such that (1) if $B=\{p\}$ then $B=\{q\}$ and (2) if $B\neq \{p\}$ then $B=\{p\}$. That's contradictory.


