In this paper Scott's reduction is discussed, which reduces any FO2 formula to Scott's Normal form. As far as I understand the reduction process explained in the paper can get you to formulas with only $\forall \forall$ and $\forall \exists$ quantification prefixes. Which has three possibilities of prefixes $\forall x \forall y $, $\forall x \exists y$ and $\exists y \forall x$. Now scott's normal form only admits $\forall x \forall y $ and $\forall x \exists y$ quantification. Then how is the explained process sound for a formula like $\exists x \forall y R(x,y) $ ?
In other words, what is the scott's reduction for $\Phi(x,y) = \exists x \forall y R(x,y)$.
My attempt on reduction according to the paper:
$ Sc( \Phi(x,y) )= \exists x P(x) \land \forall x \forall y (P(x) \leftrightarrow R(x,y)) $