Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function $$f\colon X \to Y $$ and assume that it is bijective at the level of $\mathbb{C}$-points. Moreover, assume that for every point $x$ of $X$ the differential $$df(x)\colon T_xX \to T_{f(y)}Y $$ is an isomorphism, where the tangent spaces are Zariski tangent spaces.

Can we conclude that $X$ and Y are isomorphic? Do we need any assumption on the singularities? In particular, should $Y$ be normal?

In other words, I am asking under which conditions on the singularities the implicit function theorem holds. I remember some notes by Kollar, where this issue was related with Canonical singularities, but I could not find them anymore.

Any reference or example is welcome.

thanks