suffices (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x by
  simpa [eq_bot_iff_forall, card_eq_one_iff]
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
⊢ H = ⊥ ↔ card ↥H = 1
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
⊢ (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x