have aux : ∀ ⦃C' : Set α⦄, f ∉ C' → ((M.parallelExtend e f).Circuit C' ↔ M.Circuit C') := by
  intro C' hfC'
  suffices h' : ((M.parallelExtend e f) ＼ f).Circuit C' ↔ M.Circuit C' by
    simpa [deleteElem, delete_circuit_iff, hfC'] using h'
  rw [parallelExtend_delete_eq _ hf]
α : Type u_1
inst✝ : DecidableEq α
M M' : Matroid α
e f x : α
I C : Set α
he : M.Nonloop e
hf : f ∉ M.E
hef : e ≠ f
⊢ (M.parallelExtend e f).Circuit C ↔ M.Circuit C ∨ C = {e, f} ∨ f ∈ C ∧ e ∉ C ∧ M.Circuit (insert e (C \ {f}))
α : Type u_1
inst✝ : DecidableEq α
M M' : Matroid α
e f x : α
I C : Set α
he : M.Nonloop e
hf : f ∉ M.E
hef : e ≠ f
aux : ∀ ⦃C' : Set α⦄, f ∉ C' → ((M.parallelExtend e f).Circuit C' ↔ M.Circuit C')
⊢ (M.parallelExtend e f).Circuit C ↔ M.Circuit C ∨ C = {e, f} ∨ f ∈ C ∧ e ∉ C ∧ M.Circuit (insert e (C \ {f}))