Is it true that $H(X|Y)=H(Y|X)$?

Question

I have some difficulties with the question

whether $H(X|Y)=H(Y|X)$?

From my knowledge

$I(X;Y)=H(X)-H(X|Y) = H(Y)-H(Y|X)$

so

$H(X|Y)=H(Y|X)$

only when

$H(X)=H(Y)$

The question is whether it's the last step, can I make a further assumption about the distribution of $X$ and $Y$ or $H(X)=H(Y)$ is a last step and no further conclusions.

Answer 1

$$H(X|Y)=H(Y|X)\iff H(X)=H(Y)$$