Rewriting conditional expectation

Question

Say we have random variables $B,B^1,\dots,B^k\in\mathbb{R}$. Then is it true that $$ \mathbb{E}[B|B^1,\dots,B^{k-1}] =\mathbb{E}\Big[\mathbb{E}[B|B^1,\dots,B^k]|B^1,\dots,B^{k-1}\Big]. $$ If not, then is there any way to write $f_{k-1}=\mathbb{E}[B|B^1,\dots,B^{k-1}]$ in terms of $f_{k}=\mathbb{E}[B|B^1,\dots,B^{k}]$. I'm trying to see if there is any connection between the two. I was thinking maybe $$ \mathbb{E}[B|B^1,\dots,B^{k-1}] =\mathbb{E}\Big[\mathbb{E}[B|B^1,\dots,B^k]|B^{k}\Big], $$ but this seems a bit strange to me since I don't know if law of total expectation can be applied when you are conditioning on other variables $B^1,\dots,B^{k-1}$. Thanks.