simp [hp.df.iff, hpf.df.iff, hpg.df.iff, ←imp_iff_not_or, hp.proper.iff']
L : Language
inst✝⁹ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁸ : Zero M
inst✝⁷ : One M
inst✝⁶ : Add M
inst✝⁵ : Mul M
inst✝⁴ : LT M
inst✝³ : M ⊧ₘ* 𝐏𝐀⁻
inst✝² : Structure L M
inst✝¹ : Structure.ORing L M
inst✝ : Structure.Monotone L M
Γ✝ : HierarchySymbol
k✝ : ℕ
P✝ Q : (Fin k✝ → M) → Prop
m k : ℕ
P : M → M → Prop
f g : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚫, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
pg : HSemiformula L M (k + 1) (𝚺, m + 1)
hpg : DefinedWithParam (fun v => v 0 = g fun x => v x.succ) pg
v : Fin k → M
⊢ (fun v => P (f v) (g v)) v ↔
    (Semiformula.Evalm M v id)
      ((HSemiformula.rew (Rew.substs (#0 :> fun x => #x.succ.succ)) pf).negSigma ⋎
              (HSemiformula.rew (Rew.substs (#1 :> fun x => #x.succ.succ)) pg).negSigma ⋎
                HSemiformula.rew [→ #0, #1] p.pi).all.all.val
no goals