match Γ with
| 𝚺 =>
  rcases h₁ with ⟨p₁, h₁⟩; rcases h₂ with ⟨p₂, h₂⟩
  exact ⟨p₁.negPi.or p₂, (h₁.negPi.or h₂).of_iff (fun x ↦ by simp [imp_iff_not_or])⟩
| 𝚷 =>
  rcases h₁ with ⟨p₁, h₁⟩; rcases h₂ with ⟨p₂, h₂⟩
  exact ⟨p₁.negSigma.or p₂, (h₁.negSigma.or h₂).of_iff (fun x ↦ by simp [imp_iff_not_or])⟩
| 𝚫 =>
  rcases h₁ with ⟨p₁, h₁⟩; rcases h₂ with ⟨p₂, h₂⟩; exact ⟨p₁ ⟶ p₂, h₁.imp h₂⟩
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 : ℕ
Γ : SigmaPiDelta
h₁ : Definable L (Γ.alt, m) P
h₂ : Definable L (Γ, m) Q
⊢ Definable L (Γ, m) fun v => P v → Q v
no goals