I know what 'The Five Solas' are (being of the Reformed Protestant 'school'). I know what an axiom is (established principle; self-evident truth). I know what a theorem is (proposition not self-evident but to be proved by a chain of reasoning; a truth to be established by means of accepted truths).
There is such a thing as "biblical evidence", derived from the writings of scripture, which shows that there are three basic lines of evidence God provides for humanity to know things about himself that are necessary for our salvation. Nor is it wrong to have evidence that warrants faith, because faith in God is warranted. There's nothing reprehensible about showing in what way faith is warranted. Those ways are detailed, for example, in the opening chapters of the book of Romans. Paul speaks of knowing the eternal power and invisible qualities of God from God's creation, so that those who deny them are without excuse. He also shows that the witnessed resurrection of Jesus Christ is proof that he is, indeed, the Son of God. His claims to be such while on earth are vindicated with his activities after being resurrected. He also gives further revelations to humanity via his role as the Word of God, the written record of God's words and actions being the third line of evidence. All three lines are in agreement. Protestants consider those three evidences as the basis for having come up with the Five Solas long after creation, long after Christ's time on earth, and long after the 'last word' from the Word of God was committed to writing (i.e. the Book of the Revelation).
What God has revealed of himself in creation, in the incarnation, and in his written record all came before Protestants made logical deductions about these Five Solas. They thought them through as formal statements of belief after those various events, deducing them to be self-evident in light of what had gone before. But that in no way reduces the propositions to something less than Divine Revelation, for without the three lines of God's revealing of himself to humanity, nothing about God could ever have been known with any certainty.
Needless to say, those who don't have faith in God, in the person of Christ, or in the authority of God's written word, will dismiss the Five Solas as back-to-front. As a Protestant, I would say that God-given hindsight is no bad thing. There is a logical progression to understanding the things of God, circumstances often necessitating deeper thought on already established matters of faith. The circumstances that gave rise to the Reformation in the 16th century caused deeper analysis about the fundamental (essential) points of Christian faith. It's not that the Church at the time didn't believe in God's grace as the grounds of salvation: the problem (as the Reformers saw it) was how you accessed that grace. What was the God-given foundation for knowing what he required for sinners to be saved? This is where the question's point about "the logic of salvation" comes in.
At that time, Christian people were being told to do various things to access salvation. The Reformers asked questions: could it only be secured by a sacerdotal Church system? Was it only the voice of the Church and its traditions that had to be listened to? Was it through confessing one's sins to a priest? Was it through praying to deceased saints? Was it by participating in the Mass? They found answers to all those questions in the Bible, but those who attached equal (or more) importance to Church traditions and systems than to the Bible disagreed. And therein lies the rub - where is the source of authority?
As for asking if the Five Solas are "merely true under the light of Divine Truth" - I'm astonished! There's nothing 'mere' about Divine Revelation! Or, are you using the word 'mere' in the 19th century meaning of that word, as C.S. Lewis used it in the title of his book "Mere Christianity"? If so, then that's just fine!
In conclusion, axioms and theorems apply with no trouble to algebra and geometry, but the minute beliefs of faith are examined to see whether they are axioms or theorems, there can be no agreement until first an ultimate source of authority is established. That's what the Five Solas are all about, making clear five united perspectives on salvation:
(1) The only authority for what Protestants believe is the Bible.
(2) We can only be saved through Jesus Christ and what he did.
(3) The only means of salvation is faith (in Christ) alone.
(4) The only source of salvation is God's unmerited grace.
(5) The ultimate goal of saving faith is the glory of God.
Actual answer to the actual (main) question: the Five Solas started off as a theorem and are now axiomatic.