We know $$H(X | Y) + H(Y) = H(X, Y)$$ Therefore, $$H(X | Y) \leq H(X, Y) $$ since $$ H(Y) \geq 0$$ If we expand this out, we get $$-\sum_{x,y} {p(x,y) \log p(x | y)} \leq - \sum_{x,y} {p(x,y) \log p(x, y)}$$
According to Gibbs inequality, the inequality should go in the other direction. What has gone wrong here?
The Gibbs inequality, applied to the multivariate variable $(x,y)$, tell us that $$- \sum_{x,y} {p(x,y) \log p(x, y)} \le - \sum_{x,y}{p(x,y) \log q(x, y)} $$
where $q(x, y)$ is some arbitrary (multivariate) probability function. Namely, $q(x, y)\ge 0 $ and $ \sum_{x,y} q(x,y) =1$
But $p(x\mid y)$ is not such a thing. Rather, for each fixed $y$ it's a one-dimensional (in $x$) probability function. That is, $ \sum_{x,y} p(x\mid y)= \sum_y \sum_x p(x\mid y) = \sum_y 1\ne 1$