I want to prove the following.

>For every $\Pi^0_1$ statement $\forall x\phi(x)$, where $\phi(x)$ is a $\Delta^0_1$ formula, there is $e\in\mathbb{N}$ such that $\forall x\phi(x)$ implies $W_e=PA$* and $PA+\text{$\Sigma_e$ is consistent}\vdash\forall x\phi(x)$.

The argument is inspire by an argument of Turing in his [PhD dissertation][1] (lately rephrased in [Feferman 1962][2]) to show that the transfinite progressions of adding consistency statements is sensitive on which branch we choose at the limit stage of Kleene's $\mathcal{O}$, and it is as follow.

>Fix a computable function $\sigma$ such that the range of $\sigma$ is $PA$. By recursion theorem, we can construct a partial computable function $\varphi_e$ such that

>$\varphi_e(n)=\begin{cases}\sigma(n),&\text{if $\forall x<n\phi(n)$},\\
\sigma(n)\wedge\text{$W_e$ is consistent*},&\text{o.w.}\end{cases}$

>Since $\forall x\phi(x)$ holds, $W_e=PA$. So the statement "$W_e$ is consistent" is really "$PA$ is consistent". In the next paragraph, we prove in $PA+$"$W_e$ is consistent".

>Assume $\forall x\phi(x)$ fails, than $W_e=PA+$"$W_e$ is consistent". By Goedel's second incompleteness theroem, $W_e$ is not consistent. A contradiction.

>This finishes the argument.

My question is (1) is this argument valid, or if something is missed or misunderstood? (2) is there other reference on this and related issues?

*Please forgive my abusing of notation.

  [1]: https://www.princeton.edu/turing//alan/dissertation/
  [2]: https://www.jstor.org/stable/2964649