I refer to **Stephen Cole Kleene**, *Mathematical Logic* (1967 - Dover reprint : 2002).

At pag.118 he introduces the Derived Rules for quantifiers in Predicate Calculus, beginning with $\forall$-intro : if $\Gamma\vdash A(x)$ then $\Gamma\vdash \forall x A(x)$ , provided that x is not free in $\Gamma$.

At pag.119 he comments on the necessity of the proviso:

> 
> Except for this proviso, Example 13.3 [pag.58] would apply and give $"A(x)\vdash\forall xA(x)"$. That however does not hold; for then [...] we would have $"A(x) ⊨ \forall xA(x)"$; and this is not the case.
> 
> 
> 

But in *Intro to Metamathematics* [1952] Kleene uses as Derived Rule a $\forall$-intro in the form : $"A(x) / \forall xA(x)"$.

Why this does not cause him troubles in IM ?