Transitive Replacement: $\forall x \in a \, \exists! y \ \varphi(x,y), \implies \\\exists b: Trans(b) \land a \in b \land \forall x \in a \exists! y \in b \ \varphi^b(x,y) $
where $\varphi$ is a formula that doesn't use $``b"$, and $\varphi^b$ is obtained from $\varphi$ by merely bounding all of its quantifiers by $\in b$, and $``Trans"$ signify `is transitive'
If we add this schema to Zermelo + foundation schema, then would the resulting theory prove replacement? That is, would it prove:
$\forall x \in a \, \exists! y \ \varphi(x,y), \implies \\\exists b \, \forall y \, (\exists x \in a (\varphi(x,y)) \implies y \in b) $
if it cannot prove it, then is the resulting theory equi-consistent with ZF?