Section EXTRA1.
Definition inconsistent {L : language} (Gamma : ensemble (frm L)) : Prop :=
forall p, Gamma ⊢ p.
Context {L : language}.

Lemma ExistentialE (Gamma : ensemble (frm L)) x y A B
(FRESH1 : forall p, p \in Gamma -> is_free_in_frm y p = false)
(FRESH2 : is_free_in_frm y (All_frm x A) = false)
(FRESH3 : is_free_in_frm y B = false)
(INFERS1 : Gamma ⊢ Exs_frm x A)
(INFERS2 : E.insert (subst_frm (one_subst x (Var_trm y)) A) Gamma ⊢ B)
: Gamma ⊢ B.