Here is an example I found on the web

Disjunctive Syllogism

(commutativity)

(distribution)

(contradiction)

(or-simplification)

(implication)

(De Morgan)

(associativity)

(excluded middle)

(or simplification)

Maybe it is possible to simplify hypothetical syllogism using this type of algebra?

It is easy to replace the original implications eg.

but then I am stuck with a strange looking equation?

(distributive law)

(distributiv law)

Which allows you to simplify a bit, but it's still a very big mess... Is there a simpler way? Or should I just continue on expanding using distribution?