Definition:Relativisation

Definition
Let $p$ be a well-formed formula.

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


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


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


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


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

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