Definition:Relativisation

Definition
Let $p$ be a well-formed formula of the language of set theory.

Let $A$ be a class.

The relativisation of $p$ to $A$ shall be denoted $p^A$ and shall be defined recursively on the symbols in $p$:


 * $x \in y ^A \iff x \in y$


 * $\left({\neg p}\right)^A \iff \neg p^A$


 * $\left({p \land q}\right)^A \iff \left({p^A \land q^A}\right)$


 * $\left({\forall x: P \left({x}\right)}\right)^A \iff \forall x: \left({ x \in A \implies P \left({x}\right)^A}\right)$

Thus, the relativisation of $p$ is simply the well-formed formula achieved when replacing all instances of $\forall x$ with $\forall x \in A$.