A question from Bourbaki's Theory of Sets

Question

Concerning the proof of C34 (pp. 40-41): can the argument used to prove that $$(\exists x)(\forall y)R \implies (\forall y)(\exists x)R$$ is a theorem be applied to the (false) converse?

In detail (working in $\mathscr{T}_0$): for any relation $R$ and any letter $x$, we have the theorem $R \implies (\forall y)R$ by C27. Thus $$(\forall y)(\exists x)R \implies (\exists x)R \implies (\exists x)(\forall y)R$$ is a theorem by C30 and C31.

Answer 1

There is no error...

C27. says :

if $R$ is a theorem, then $(\forall x) \ R$ is a theorem.

In more "modern" notation :

if $\vdash R$, then $\vdash (\forall x) \ R$.

This is not the same as : $\vdash R \to (\forall x) \ R$, that is not valid.