simp only [circuit_iff_dep_forall_diff_singleton_indep, uniqueBaseOn_dep_iff,
  uniqueBaseOn_indep_iff', subset_inter_iff, diff_singleton_subset_iff, mem_diff]
α : Type u_1
M : Matroid α
C C' I X K C₁ C₂ R : Set α
e f x y : α
E D : Set α
⊢ (uniqueBaseOn I E).Circuit C ↔ ∃ e ∈ E \ I, C = {e}
α : Type u_1
M : Matroid α
C C' I X K C₁ C₂ R : Set α
e f x y : α
E D : Set α
⊢ ((C.Nonempty ∧ ¬C ⊆ I ∧ C ⊆ E) ∧ ∀ e ∈ C, C ⊆ insert e I ∧ C ⊆ insert e E) ↔ ∃ e, (e ∈ E ∧ e ∉ I) ∧ C = {e}