To prove the equivalence to the Lebesgue measures

Question

Let $\pi$ be a probability measure on $\mathbb{R}^2$ absolutely continuous w.r.t. the Lebesgue measure on $\mathbb{R}^2$. Denote by $f$ its density function, and by $\pi_x, \pi_y$ the two marginal distributions on $\mathbb R$. We know that \begin{equation} {\rm Leb}\times \pi_y \{(x,y)|~f(x, y)=0\}=0 =\pi_x \times {\rm Leb} \{(x,y)|~f(x, y)=0\}. \end{equation} Is it true that $\pi$ is equivalent to the Lebesgue measure, or is there a counterexample? Thanks!