Let x0 of type ι be given.
Apply and3I with
∀ x1 . x1 ∈ x0 ⟶ x1 ∈ x0,
∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ x1 = x2 ⟶ x1 = x2,
∀ x1 . x1 ∈ x0 ⟶ ∃ x2 . and (x2 ∈ x0) (x2 = x1) leaving 3 subgoals.
Let x1 of type ι be given.
Assume H0: x1 ∈ x0.
The subproof is completed by applying H0.
Let x1 of type ι be given.
Assume H0: x1 ∈ x0.
Let x2 of type ι be given.
Assume H1: x2 ∈ x0.
Assume H2: x1 = x2.
The subproof is completed by applying H2.
Let x1 of type ι be given.
Assume H0: x1 ∈ x0.
Let x2 of type ο be given.
Assume H1:
∀ x3 . and (x3 ∈ x0) (x3 = x1) ⟶ x2.
Apply H1 with
x1.
Apply andI with
x1 ∈ x0,
x1 = x1 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ι → ι → ο be given.
Assume H2: x3 x1 x1.
The subproof is completed by applying H2.