Local Open Scope logic_base.
Local Open Scope syntax.
Section DerivableRulesFromPatterns.
Context {L: Language}
{minL: MinimunLanguage L}
{pL: PropositionalLanguage L}
{Gamma: ProofTheory L}
{minAX: MinimunAxiomatization L Gamma}
{ipGamma: IntuitionisticPropositionalLogic L Gamma}
{prodp: expr -> expr -> expr}.

Lemma prodp_comm {Comm: Commutativity L Gamma prodp}: forall x y,
|-- prodp x y <--> prodp y x.