∀ (x : SetTheory.Game) (motive : (∃ y, ⟦y⟧ = x ∧ y.birthday = x.birthday) → Prop)
  (x_1 : ∃ y, ⟦y⟧ = x ∧ y.birthday = x.birthday),
  (∀ (y : SetTheory.PGame) (hy₁ : ⟦y⟧ = x) (hy₂ : y.birthday = x.birthday), motive ⋯) → motive x_1