There are a few things to unpick, here.
First, there's a difference between provability in a formal system, and "truth", which is a question of the relationship between language acts and facts. The statement "Juh mapple Neele" is neither true nor false, but nonsense, unless it is recognised as a poorly pronounced version of the French phrase Je m'appele Niel. The existence of a correspondance between the utterances and facts is necessary to have a meaningful notion of spoken truth. Similarly, your string
$$$$$ which is obtainable by transformations from
$$$$ is not 'true', except if you provide some correspondance between it and some model. That is to say, you haven't provided any thing for
$$$$$ to be true of, nor a notion of what it would mean for it to be true.
Of course, French — and English, and all other natural languages — don't have a formal specification of how they correspond to reality. We more or less learn the relationship between the language and reality by correlations, and by depending on one another to mostly co-operate in reinforcing the conventional correspondance between language and reality. If we tried to describe the correspondance, it would just be with more language; just as we tend to use mathematics to make statements about meta-mathematics, and sometimes try to describe logic by boolean algebras. Our formal use of logic is a carefully honed skill of associations, a refined sort of activity of the same sort as natural language usage is, sharpening that more common skill to scalpel sharpness and rigidity.
Having said this, how do we know that we reason correctly, logically? Lewis Caroll (of Alice in Wonderland fame) wrote precisely on this subject about logical regress in certainty of provability in "What the Tortoise said to Achilles". The moral of the story is: if you are sufficiently skeptical of how to apply logical rules of inference, then any logical conclusion is impossible to reach, because you cannot prove that you have correctly proven something without putting the skepticism about proof simply at one more remove. The rules of inference are best practises for obtaining transformations of sentences in such a way that it minimally increases the error of the statement. In its role in formal derivations, it may be regarded as a sort of dance step — a simple move to be mastered as part of more complicated coreography of precise motions to convey a message, which is interpreted by others who know the correspondance of the formal system to other systems of thinking and imagining, sometimes consisting of physical reality or a caricature of it.
If you are not confident in your ability to make simple motions, e.g. to make simple speech acts, then no linguistic argument will convince you that language has any correspondance to reality; and the same is true of formal logic. What formal logic allows us to do is to check whether an argument has any steps which we don't have very much confidence in. That is, if we are sufficiently skeptical, the best we can do is say that we cannot find obvious flaws in the reasoning; the complementary case being when we can find obvious flaws, as in an invalid derivation. Of course, it is possible to make mistakes when identifying a step of reasoning as invalid, but at least it provides us with the opportunity to try to demonstrate it's falsehood by attempting to imagine a refutation of the statement being asserted by focusing on the consequences of the mis-step. Then we may try to bring our resources to bear, if we are worried that we have reasoned poorly, to ask whether the statement to be proved or the imagined refutation seem more coherent with our models.
Of course, as respects captial-T Truth, one might say that any claim to have access to absolute truth represents an act of hubris: that the world is necessarily so simple as to be completely comprehended by a mind handily contained within a cubic foot of space. To anyone familiar with the phenomena of turbulence and quantum mechanics, or the mathematical construction of fractals or strange attractors, or indeed the Halting problem and the P vs. NP problem — in short, the mathematical discoveries or problems of the 20th century — the idea that absolute capital-T Truth is something which is easily accessible even from premisses of analytical philosophy is laughable, unless the statement is so vague as to communicate almost nothing of the world's complexity, or so elaborate that it exists only as a script which was deliberately and pains-takingly recorded; the knowledge consisting more in the ability to decipher and recite the record as a sort of ritual for the purpose of undertaking some act in the world of which the record somehow represents an incomplete but possibly helpful sketch. Formal logic is a school of such ritual performance, which describes a particularly reliable way to formulate and perform such rituals.
Logic is not a guarantor of truth, but merely (!) a very powerful tool for precise argumentation, and one which has a very good record of reliability, even if many people find it a difficult tool to wield carefully. As with any tool, it is only as reliable as the person who uses it. But somehow, as with the tools of art and of construction, we find that we can often recognise when someone has used it skilfully, and aspire to ideals of not only precise but elegant and artful wielding of this tool — which after all is a form of language.
The role of formal logic is, in short, a form of exercise in which to try to reason carefully, and to learn what careful reasoning looks like. But it cannot teach one to reason, nor guarantee good reasoning, without having some basic skill already.