Section More_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 subSubFormula :
forall (f : Formula) (v1 v2 : nat) (s1 s2 : Term),
v1 <> v2 ->
~ In v1 (freeVarTerm L s2) ->
forall T : System,
SysPrf T
(iffH (substituteFormula L (substituteFormula L f v1 s1) v2 s2)
(substituteFormula L (substituteFormula L f v2 s2) v1
(substituteTerm L s1 v2 s2))).