∀ {α β : Type u_1} {S : Set (α ⊕ β → ℕ)} (v : α → ℕ) (motive : {v | ∃ x, Sum.elim v x ∈ S} v → Prop)
  (x : {v | ∃ x, Sum.elim v x ∈ S} v), (∀ (x : β → ℕ) (hx : Sum.elim v x ∈ S), motive ⋯) → motive x