Derivation of implicational propositional axioms
The natural way to define your addition operation is as set intersection. The value of a single axiom is the set of all models satisfying the axiom. The sum of several axioms is then the set of all models satisfying all the axioms.

A similar definition is $\mathbf{A} + \mathbf{B} = \mathbf{A} \land \mathbf{B}$, but then you need to interpret equality as logical equivalence. The definitions then become completely equivalent.

However, you can't "cancel" axioms. If you know that $\mathbf{A}+\mathbf{B} = \mathbf{A}+\mathbf{C}$ then all you can conclude is that in the presence of $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ are equivalent. As an example, let $\mathbf{A}$ be $x=0$, let $\mathbf{B}$ be $y=1$, and let $\mathbf{C}$ be $x+y=1$. In the presence of $\mathbf{A}$, the axioms $\mathbf{B}$ and $\mathbf{C}$ are equivalent, but in general they aren't.

If are are really keen on the cancellation law, you can always take formal sums. It is then true that $\mathbf{A}+\mathbf{B} = \mathbf{A}+\mathbf{C}$ implies $\mathbf{B}=\mathbf{C}$, but at the cost of $\mathbf{A}+\mathbf{B}$ having no other meaning than the formal sum of $\mathbf{A}$ and $\mathbf{B}$.