Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Assume H0: x1 ∈ x0.
Assume H1: ∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0.
Assume H2: ∀ x3 . x3 ∈ x0 ⟶ x2 x3 = x1 ⟶ ∀ x4 : ο . x4.
Assume H3: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4.
Assume H4: ∀ x3 : ι → ο . x3 x1 ⟶ (∀ x4 . x3 x4 ⟶ x3 (x2 x4)) ⟶ ∀ x4 . x4 ∈ x0 ⟶ x3 x4.
Apply and5I with
x1 ∈ x0,
∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0,
∀ x3 . x3 ∈ x0 ⟶ x2 x3 = x1 ⟶ ∀ x4 : ο . x4,
∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4,
∀ x3 : ι → ο . x3 x1 ⟶ (∀ x4 . x3 x4 ⟶ x3 (x2 x4)) ⟶ ∀ x4 . x4 ∈ x0 ⟶ x3 x4 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H4.