The general question is
Can I define an axiomatic system and prove theorems using Mathematica?
The more concrete one is about Boolean algebra.
I consider this axiomatic Boolean algebra system (wiki).
How can I
define the six (or twelve) axioms there in Mathematica and
then let Mathematica prove theorems in the user-defined axiomatic system, instead of the built-in system in Mathematica?
(Theorems like: De Morgan's law ($\lnot(a \lor b) \equiv \lnot a \land \lnot b$) or the easier (maybe harder) ones such as $\lnot(\lnot a) \equiv a$ and $\lnot 0 \equiv 1$.
You are not limited to Boolean algebra. You can show your skills in any fields you are good at.