Section Logic_Rules.
Variable L : Language.
Let Formula := Formula L.
Let Formulas := Formulas L.
Let System := System L.
Let Term := Term L.
Let Terms := Terms L.
Let var := var L.
Let apply := apply L.
Let equal := equal L.
Let atomic := atomic L.
Let impH := impH L.
Let notH := notH L.
Let forallH := forallH L.
Let orH := orH L.
Let andH := andH L.
Let existH := existH L.
Let iffH := iffH L.
Let ifThenElseH := ifThenElseH L.
Let Prf := Prf L.
Let SysPrf := SysPrf L.

Lemma existSys :
forall (T : System) (f g : Formula) (v : nat),
~ In_freeVarSys L v T ->
~ In v (freeVarFormula L g) ->
SysPrf (Ensembles.Add _ T f) g -> SysPrf (Ensembles.Add _ T (existH v f)) g.