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It follows from the Reflection Principle that every ordinal definable set is ordinal definable by a $\Sigma_2$-formula in the language of set theory. Indeed, if $A = \{x : \phi(x,\bar\alpha)\}$, then $$\exists\gamma\\,(\gamma \in \mathrm{Ord} \land \bar\alpha \in \gamma \land \forall x\\,(x \in A \leftrightarrow x \in V_\gamma \land V_\gamma \vDash \phi(x,\bar\alpha))).$$ Therefore, there is some $\gamma \in \mathrm{Ord}$ such that $$x \in A \leftrightarrow \exists U\\,(U = V_\gamma \land x,\bar\alpha \in U \land U \vDash \phi(x,\bar\alpha)),$$ which is $\Sigma_2$ since $U = V_\gamma$ is $\Pi_1$.

It follows from the Reflection Principle that every ordinal definable set is ordinal definable by a $\Sigma_2$-formula in the language of set theory. Indeed, if $A = \{x : \phi(x,\bar\alpha)\}$, then $$\exists\gamma(\gamma \in \mathrm{Ord} \land \bar\alpha \in \gamma \land \forall x(x \in A \leftrightarrow x \in V_\gamma \land V_\gamma \vDash \phi(x,\bar\alpha))).$$ Therefore, there is some $\gamma \in \mathrm{Ord}$ such that $$x \in A \leftrightarrow \exists U(U = V_\gamma \land x,\bar\alpha \in U \land U \vDash \phi(x,\bar\alpha)),$$ which is $\Sigma_2$ since $U = V_\gamma$ is $\Pi_1$.

It follows from the Reflection Principle that every ordinal definable set is ordinal definable by a $\Sigma_2$-formula in the language of set theory. Indeed, if $A = \{x : \phi(x,\alpha)\}$, then $$\exists\gamma\\,(\gamma \in \mathrm{Ord} \land \forall x\\,(x \in A \leftrightarrow x \in V_\gamma \land V_\gamma \vDash \phi(x,\alpha))).$$ Therefore, there is some $\gamma \in \mathrm{Ord}$ such that $$x \in A \leftrightarrow \exists U\\,(U = V_\gamma \land x,\alpha \in U \land U \vDash \phi(x,\alpha)),$$ which is $\Sigma_2$ since $U = V_\gamma$ is $\Pi_1$.

It follows from the Reflection Principle that every ordinal definable set is ordinal definable by a $\Sigma_2$-formula in the language of set theory. Indeed, if $A = \{x : \phi(x,\bar\alpha)\}$, then $$\exists\gamma\\,(\gamma \in \mathrm{Ord} \land \bar\alpha \in \gamma \land \forall x\\,(x \in A \leftrightarrow x \in V_\gamma \land V_\gamma \vDash \phi(x,\bar\alpha))).$$ Therefore, there is some $\gamma \in \mathrm{Ord}$ such that $$x \in A \leftrightarrow \exists U\\,(U = V_\gamma \land x,\bar\alpha \in U \land U \vDash \phi(x,\bar\alpha)),$$ which is $\Sigma_2$ since $U = V_\gamma$ is $\Pi_1$.

It follows from the Reflection Principle that every ordinal definable set is ordinal definable by a $\Sigma_2$-formula in the language of set theory. Indeed, if $A = \{x : \phi(x,\alpha)\}$, then $$\exists\gamma\\,(\gamma \in \mathrm{Ord} \land \forall x\\,(x \in A \leftrightarrow x \in V_\gamma \land V_\gamma \vDash \phi(x,\alpha))),$$ which is $\Sigma_2$ since the satisfaction relation for $V_\gamma$ and the relations $x \in V_\gamma$, $\gamma \in \mathrm{Ord}$, are all $\Delta_1$.

It follows from the Reflection Principle that every ordinal definable set is ordinal definable by a $\Sigma_2$-formula in the language of set theory. Indeed, if $A = \{x : \phi(x,\alpha)\}$, then $$\exists\gamma\\,(\gamma \in \mathrm{Ord} \land \forall x\\,(x \in A \leftrightarrow x \in V_\gamma \land V_\gamma \vDash \phi(x,\alpha))).$$ Therefore, there is some $\gamma \in \mathrm{Ord}$ such that $$x \in A \leftrightarrow \exists U\\,(U = V_\gamma \land x,\alpha \in U \land U \vDash \phi(x,\alpha)),$$ which is $\Sigma_2$ since $U = V_\gamma$ is $\Pi_1$.