Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Let x3 of type ο be given.
Assume H1:
(∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x1) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5) ⟶ (∀ x4 . x4 ∈ x1 ⟶ ∃ x5 . and (x5 ∈ x0) (x2 x5 = x4)) ⟶ x3.
Apply H0 with
x3.
Assume H2:
and (∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x1) (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5).
Apply H2 with
(∀ x4 . x4 ∈ x1 ⟶ ∃ x5 . and (x5 ∈ x0) (x2 x5 = x4)) ⟶ x3.
Assume H3: ∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x1.
Assume H4: ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5.
Assume H5:
∀ x4 . x4 ∈ x1 ⟶ ∃ x5 . and (x5 ∈ x0) (x2 x5 = x4).
Apply H1 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.