Is it true that $H(Y|X_1,\dots,X_n)=H(X_1,\dots,X_n,Y)-H(X_1,\dots,X_n)$?

Question

Suppose $X_1,\dots,X_n,Y$ are random variables. Is it true that the conditional entropy can be expressed as the difference between the joint entropy of all variables and the joint entropy of only $X$ variables? That is: $$H(Y|X_1,\dots,X_n)=H(X_1,\dots,X_n,Y)-H(X_1,\dots,X_n)$$

Answer 1

Yes it is. Consider the big tuple $(X_1,\cdots ,X_n)$ as one random variable $Z$. Hence $$H(Y|Z)=H(Z,Y)-H(Z)$$which is indeed true for any two random variables $Y,Z$.