Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι be given.
Apply iffI with
x2 ∈ Pi x0 x1,
and (∀ x3 . x3 ∈ x2 ⟶ and (pair_p x3) (ap x3 0 ∈ x0)) (∀ x3 . x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3) leaving 2 subgoals.
The subproof is completed by applying PiE with x0, x1, x2.
Assume H0:
and (∀ x3 . x3 ∈ x2 ⟶ and (pair_p x3) (ap x3 0 ∈ x0)) (∀ x3 . x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3).
Apply H0 with
x2 ∈ Pi x0 x1.
Assume H1:
∀ x3 . x3 ∈ x2 ⟶ and (pair_p x3) (ap x3 0 ∈ x0).
Assume H2:
∀ x3 . x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3.
Apply PiI with
x0,
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.