rw [← dual_dual M, ← dual_delete_dual_eq_contract, dual_base_iff', dual_base_iff',
  delete_base_iff, dual_dual, delete_ground, diff_diff, union_comm, union_subset_iff,
  subset_diff, ← and_assoc, and_congr_left_iff, dual_ground, and_iff_left hIE, and_congr_left_iff]
case intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I J R B X Y Z K C C₁ C₂ : Set α
hIE : I ⊆ M.E
B₀ : Set α
hB₀ : M✶.Base B₀
hfk : B₀ ⊆ M✶.E \ I
⊢ (M ／ I).Base B ↔ M.Base (B ∪ I) ∧ Disjoint B I
case intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I J R B X Y Z K C C₁ C₂ : Set α
hIE : I ⊆ M.E
B₀ : Set α
hB₀ : M✶.Base B₀
hfk : B₀ ⊆ M✶.E \ I
⊢ Disjoint B I → B ⊆ M.E → (M✶.Basis (M.E \ (B ∪ I)) (M.E \ I) ↔ M✶.Base (M.E \ (B ∪ I)))