Fixpoint addForalls (m n : nat) (f : Formula) {struct n} : Formula :=
  match n with
  | O => f
  | S n' => forallH (n' + m) (addForalls m n' f)
  end.

Lemma reduceAddForalls : forall (n m : nat) (A B : Formula), SysPrf NN (iffH A B) -> SysPrf NN (iffH (addForalls m n A) (addForalls m n B)).