Let x0 of type ι be given.
Apply and3I with
∀ x1 . prim1 x1 x0 ⟶ prim1 x1 x0,
∀ x1 . prim1 x1 x0 ⟶ ∀ x2 . prim1 x2 x0 ⟶ x1 = x2 ⟶ x1 = x2,
∀ x1 . prim1 x1 x0 ⟶ ∃ x2 . and (prim1 x2 x0) (x2 = x1) leaving 3 subgoals.
Let x1 of type ι be given.
The subproof is completed by applying H0.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H2: x1 = x2.
The subproof is completed by applying H2.
Let x1 of type ι be given.
Let x2 of type ο be given.
Assume H1:
∀ x3 . and (prim1 x3 x0) (x3 = x1) ⟶ x2.
Apply H1 with
x1.
Apply andI with
prim1 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.