simp only [def_exists_]
case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
    (((forall_ v (P.iff_ Q)).imp_ ((exists_ v Q).imp_ (exists_ v P))).imp_
      (((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))).imp_
        ((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).iff_ (exists_ v Q)))))
case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
    (((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_)).imp_
      (((forall_ v (P.iff_ Q)).imp_ ((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_)).imp_
        ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P.not_).not_.iff_ (forall_ v Q.not_).not_))))