Definition dnQ {b} (phi : form b) : form b := (phi --> Q) --> Q.
Fixpoint trans {b} (phi : form b) : form b :=
match phi with
| Impl phi1 phi2 => Impl (trans phi1) (trans phi2)
| All x phi => All x (trans phi)
| Pr s t => dnQ (Pr s t)
| Q =>  Q
| Fal => Q
end.

Lemma app1 b psi1 psi2 A :
(psi1 --> psi2 :: A) ⊢I psi1 -> (psi1 --> psi2 :: A) ⊢I psi2.