I'm self-studying Robert Causey's Logic, Sets and Recursion (2nd edition), I totally can't understand the statement of metatheorem 4-13, which said
I think the reason I don't understand is due to two reasons. One is I missed in the verbal description about the hypothesis of $\phi,~\mathbf{I}$ and $\mathbf{J}$, the other is that English is not my native language, so I think I don't quite get the meaning of this sentence "where $\psi$ has occurrences of $\kappa$, but not necessarily all occurrences of $\kappa$ are replaced by $\beta$" and the usage of "as it" in the sixth line. Also the definition of $\mathbf{I}$ and $\mathbf{J}$. Can you help me by some concrete examples, or states in another way?
Maybe an example helps. Suppose $\psi$ is $P(\kappa,\alpha) \vee Q(\kappa)$ and $\phi$ is $P(\beta,\alpha) \vee Q(\kappa)$. Suppose $\mathbf{I}$ and $\mathbf{J}$ interpret $P, Q, \alpha$, and $\kappa$ the same way. In addition $\mathbf{J}$ gives the same interpretation to $\beta$ and $\kappa$. Then the truth value of $\psi$ under $\mathbf{I}$ is the same as the truth value of $\phi$ under $\mathbf{J}$.
The "where" refers to the fact that the occurrences of $\beta$ are in places where the other formula has $\kappa$s.