Ah yes, this is probably a more intuitive way of doing things than in my comment (though of course we prove different things).⁰ I should add that for those interested, the SEP page on algebraic propositional logic is quite interesting, although I regrettably have not actually gotten round to working through it.¹

0: https://news.ycombinator.com/item?id=28756727

1: https://plato.stanford.edu/entries/logic-algebraic-propositi...