Can modern, Democrat Liberal ideology be proven logically, ie, does it extend rationally from logical axioms, or does it have other means of axiomatization.
A political theory or philosophy is more or less equivalent to an induction. In other words, a theorist will begin with some insights about the nature of humans in society and will generalize from those insights to an abstract universal system.
Inductions aren't proved in the sense of deductive logic; inductions are inferred, and tested.
Specifically for Liberalism, the early Liberal philosophers had certain insights about human nature — a capacity for reason, a necessary investment in labor for continued survival, an intrinsic political equality, etc — and inferred various principled systems of governance, economics, and society that maximized those characteristics and minimized certain weaknesses. These principles were then implemented within various democratic republics. But the success of those republics is not guaranteed (as it would be if they were produced through mere deduction). Each form of republic is effectively an experimental model that is being tested through its real-time practice.
With luck, someone is paying attention to which experiments work and which don't, and learning How to improve the Liberal project. If not, well... please pick which Mad Max movie you'd like to live in.
It can't be "proven" in any sense, but liberalism is, in theory, an extension of certain "self-evident" axioms, equality chief among them. Obviously, equality is not an observable fact, quite the opposite, and so it may be deemed purely axiomatic.
The logical extension, however, is not necessarily liberalism. Hobbes begins with the assertion that all men are "equally" capable of killing each other and uses this to "prove" the necessity of absolute executive authority.
As an outgrowth of the Enlightenment, theories of liberalism, from Locke to Rawls, are generally developed through axioms and "rational" arguments, as opposed to dogma, faith, custom, or moral ideals.
But this is theory, not proof. The existence of many other forms of government is empirical proof that there is nothing logically necessary about any one type of government.