If true for the general element, then true for all. What's this? In mathematics often (always) one proves that a property is true for the general element of a set. From that, one can say that that property is true for all the elements of that set.
Is that a principle of logic or set theory? What is it called? When I make this kind of reasoning, what should I think of?
Thanks,
Luca

The Universal Generalization basically shows that if a particular property holds for an arbitrary element of a set, then it holds generally for any element of it. The reason this rule is valid is because we assume absolutely nothing about the element $x$ we pick up.
From Velleman's How to Prove It (2006), p.108:

To prove a goal of the form ∀x P(x):
Let $x$ stand for an arbitrary object and prove $P(x)$. The letter $x$ must be a
new variable in the proof. If $x$ is already being used in the proof to stand for
something, then you must choose an unused variable, say $y$, to stand for the
arbitrary object, and prove $P(y)$.

Hence the chosen element must be arbitrary, that is, we express this by stating that the letter chosen to denote it must have no previous ocurrences in the proof (a new variable).
