. intro h Γ Δ hΓ hΔ;
  simp_all;
  have : 𝓓 ⊬! Γ.conj' ⟶ p ⋎ (Δ.remove p).disj' := h hΓ (by
    intro q hq;
    have := by simpa using hΔ q $ List.mem_of_mem_remove hq;
    cases this with
    | inl h => simpa [h] using List.mem_remove_iff.mp hq;
    | inr h => assumption;
  );
  by_contra hC;
  have : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ (Δ.remove p).disj' := imp_trans! hC $ forthback_disj'_remove;
  contradiction;
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj') →
    (𝓓)-Consistent (T, insert p U)
no goals