Derivation rules kind of look like fractions, don't they? Let's try applying our usual calculation rules for fractions and see what we get.

× =

Hmm. If we interpret A B to mean A∧B, that makes sense. By multiplying two rules, we get another valid rule, for if we know that A→B and C→D, then clearly A∧C→B∧D.

Let's try addition.

+ =

Now, this one is less obvious. Let us interpret A+B to mean A∨B. Can we deduce (A∧D)∨(B∧C)→(B∧D) from A→B and C→D? Indeed we can:

So derivation rules do, in fact, behave a bit like fractions, and by applying some rules from high-school algebra, we can create new, valid rules from given derivations. Weird stuff. 😉