Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Apply unknownprop_283f273b4f28592b1422a49659c7307e57b55f3b51858c4b8b39e82c747a5bc9 with
λ x3 x4 : ι → ι → (ι → ι) → ο . x4 x0 x1 x2 ⟶ ∀ x5 : ο . ((∀ x6 . In x6 x0 ⟶ In (x2 x6) x1) ⟶ (∀ x6 . In x6 x0 ⟶ ∀ x7 . In x7 x0 ⟶ x2 x6 = x2 x7 ⟶ x6 = x7) ⟶ x5) ⟶ x5.
Assume H0:
(λ x3 x4 . λ x5 : ι → ι . and (∀ x6 . In x6 x3 ⟶ In (x5 x6) x4) (∀ x6 . In x6 x3 ⟶ ∀ x7 . In x7 x3 ⟶ x5 x6 = x5 x7 ⟶ x6 = x7)) x0 x1 x2.
Apply andE with
∀ x3 . In x3 x0 ⟶ In (x2 x3) x1,
∀ x3 . In x3 x0 ⟶ ∀ x4 . In x4 x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4.
The subproof is completed by applying H0.