I'm studying Set Theory on my own and I have a question about a proof. The book I am reading wants to prove $ZF \vdash (AC)^L$ in order to prove the relative consistency of AC from ZF. I have no problems with the proof of the following theorem:

1) $ZF \vdash (V = L)^L.$

Here comes the problem: my book is a bit concise and I am not sure if the following proof of mine is correct. It would be very kind if you could check it:

We have a formula $\Phi$ which is $\exists \xi \psi$ where $\psi$ is $\Delta^{ZF}_1$ (I do not know if you are familiar with this $\psi$ but it should be the canonical well-ordering of $L$), so from 1 we have $$ZF \vdash \forall x \in L (\Phi \text{ Well orders } x)^L$$ which is equivalent to $$ZF \vdash \forall x \in L (\Phi^L \text{ Well orders } x)$$ (because the formula $\Phi^L \text{ Well orders } x$ is $\Delta_1^{ZF}$)

which is equivalent to $$ZF \vdash \forall x \in L (\Phi \text{ Well orders } x)$$ (because $\Phi$ is $\exists \xi \psi$ and all the ordinals are in $L$)

which is $$ZF \vdash WOT^L$$ (where WOT is the Well Ordering Theorem)

But also $ZF \vdash WOT \leftrightarrow AC$ so $ZF \vdash AC^L$.

QED?

Note: The last passage works because we have $ZF \vdash WOT^L \leftrightarrow AC^L$ because $WOT \leftrightarrow AC$ is $\Delta_1^{ZF}$.