Are $\sf ZF+Con(\sf ZF),\sf ZFC+Con(\sf ZF)$ equiconsistent? It's very well known that (over base theory being $\sf ZF$) theories $\sf ZF$ and $\sf ZFC$ are equiconsistent. Is the same known to be true about $\sf ZF+Con(\sf ZF)$ and $\sf ZFC+Con(\sf ZF)$? How about extending this to higher levels of consistency hierarchy, namely, are $\sf ZF_1+Con(ZF_2+Con(...+Con(ZF_n)...))$ all equiconsistent, where $\sf ZF_i\in\\{ZF,ZFC\\}$? I looked a little on the web, but I could find no references. Thanks in advance.

Yes. Given an arithmetic statement $\varphi$, $\sf ZF+\varphi$ is equiconsistent with $\sf ZFC+\varphi$.

To see why, first note that one implies the other trivially. If $\sf ZFC+\varphi$ holds, then $\sf ZF+\varphi$ is certainly true.

In the other direction if $M$ is a model of $\sf ZF+\varphi$, then $L^M$ is a model of $\sf ZFC$. But as far as $M$ is concerned, $M$ and $L^M$ have the same natural numbers and the same arithmetic structure. Therefore they have the same theory.

Because the statement "$X$ is a structure for the language $\cal L$, and $\varphi$ is a statement true in $X$" is a $\Delta_0$ statement (with parameters $X$, $\cal L$ and $\varphi$). And so if the in $L^M$ an arithmetic statement $\varphi$ holds, then it must hold in $M$ as well.