Section DEDUCTION_THEOREM.
End DEDUCTION_THEOREM.

Lemma Gen_preserves_truth (x : ivar) (q : frm L) (ps : list (frm L)) (Gamma : ensemble (frm L))
(NOT_FREE : forall p, In p ps -> is_not_free_in_frm x p)
(ENTAILS1 : forall Gamma, E.fromList ps \subseteq Gamma -> Gamma ⊨ q)
(INCL : E.fromList ps \subseteq Gamma)
: Gamma ⊨ All_frm x q.