rw [Set.mem_inter_iff, SetLike.mem_coe, SetLike.mem_coe] at hx
case refine'_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ ↑A ∩ ↑B
⊢ x = 1
case refine'_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x = 1