Section TruthLemma.
Context {L: Language}
{minL: MinimumLanguage L}
{pL: PropositionalLanguage L}
{Gamma: Derivable L}
{bSC: BasicSequentCalculus L Gamma}
{minSC: MinimumSequentCalculus L Gamma}
{ipSC: IntuitionisticPropositionalSequentCalculus L Gamma}
{cpSC: ClassicalPropositionalSequentCalculus L Gamma}
{MD: Model}
{kMD: KripkeModel MD}
{M: Kmodel}
{SM: Semantics L MD}
{tminSM: TrivialMinimumSemantics L MD SM}
{tpSM: TrivialPropositionalSemantics L MD SM}.
Context (cP: context -> Prop)
(rel: bijection (Kworlds M) (sig cP)).
Context (AL_MC: at_least (maximal consistent) cP).

Lemma truth_lemma_impp
(x y: expr)
(IHx: forall m Phi, rel m Phi -> (KRIPKE: M, m |= x <-> proj1_sig Phi x))
(IHy: forall m Phi, rel m Phi -> (KRIPKE: M, m |= y <-> proj1_sig Phi y)):
forall m Phi, rel m Phi -> (KRIPKE: M, m |= x --> y <-> proj1_sig Phi (x --> y)).