I'm currently reading Ethan D. Bloch's book "Proofs and Fundamentals: A First Course in Abstract Mathematics". I know this is not a good book for learning logic, but this is not why I'm reading it. However, I came across some claims that are really bugging me and I would appreciate if someone more knowledgeable in formal logic could provide me some clarification. The claims in question are the following (section 1.4, pp. 28-29, emphasis mine):
This sort of justification, often referred to by logicians as a derivation, is a chain of statements connected by meta-statements (namely, the justifications for each line). If an argument has a derivation, we say that the argument is derivable. [...] We now face an important question: given an argument, we have two notions of whether the argument works, which are that it is or is not valid, and that it is or is not derivable. The former notion involves checking truth values (which is done with truth tables), the latter constructing a chain of statements linked by rules of inference. What is the relation between these two approaches? Though it is not at all obvious, nor easy to prove, it turns out quite remarkably that these two approaches, while different in nature, always yield the same result. That is, an argument is valid if and only if it is derivable. Hence, if we want to show that a given argument is valid, it will suffice to show that it is derivable, and vice versa. The equivalence of these two approaches is a major result in logic. That validity implies derivability is often referred to as the “Completeness Theorem,” and that derivability implies validity is often referred to as the “Soundness Theorem” or “Correctness Theorem.”
What I understand about the above text is that it's not obvious or easy to see that derivation and validity are equivalent notions. That is, someone must've thought that it would be possible, at least in theory, that we could find a derivation for an invalid argument, thus the need for the so called "Soundness Theorem". However (and this is the part that I'm struggling to understand), if the rules of inference are already proved to be valid arguments and we use them to come up with a derivation for a certain argument (i.e., we can "transform" the premises into the conclusion), then, certainly, the argument in question must be valid, right? And this is what's bugging me; contrary for what the author is asserting in the above text, I find it easy to see a relation between these two notions, so there's certainly something that I'm misunderstanding.
I know it's impossible to come up with an example of a derivation for an invalid argument, but can somebody help me better understand how the notion of validity is (or could be) independent from the notion of derivation.
PS: I also consulted these two questions on the web, but I'm still struggling to understand the separation between these two notions: