Finite transitive counterexample for power-set equality ($M\models 3=\mathcal{P}(2),V_4/\models 3=\mathcal{P}(2)$)

Let $M=4$ be a Finite von Neumann ordinal and $N=V_4$. With parameters $x=3$, $y=2$, the transitive structure $M$ satisfies the first-order power-set specification, since its available subsets of $2$ are exactly $0,1,2$. In $N$, the additional subset $\{1\}$ makes the same specification false. Bounded logical formulas are absolute between transitive sets by induction, so existential logical formulas with bounded matrices are upward absolute. This counterexample therefore rules out an unrestricted logical equivalence of power-set equality with a ${\Sigma }_1$ logical formula.