I have been reading Kleene's "Introduction to Metamathematics" Chapter 5 Section 24 where it is stated that $A(x) \vdash \forall xA(x)$ is a deduction rule. I was wondering on the interpretation of this rule and its analog in informal mathematics. For me, I interpret it as that if some statement $A$ for some variable $x$ is true then it is true for all $x$, but it does not make sense for me because, for example, if $A$ is " is prime " then for some number it is true that it is prime but for all numbers that is not true.
I would appreciate help and any comments about this!