Questions on the functionality of Mathematica for Boolean computation.

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Can I define an axiomatic (Boolean algebra) system and prove theorems using Mathematica?

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 ...
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What are the steps using boolean algebra for transforming one equation into another?

I have a special case where I have two equations, which I have solved in a better and quicker way, but I need to solve with Boolean algebra. Using the laws of Boolean algebra, I have to prove that ...