Let x0 of type (ι → ι → ι) → ο be given.
Assume H0: ∃ x1 : ι → ι → ι . x0 x1.
Assume H1: ∀ x1 x2 : ι → ι → ι . x0 x1 ⟶ x0 x2 ⟶ x1 = x2.
Apply H0 with
x0 (Descr_iii x0).
Let x1 of type ι → ι → ι be given.
Assume H2: x0 x1.
Apply functional extensionality with
x1,
Descr_iii x0.
Let x2 of type ι be given.
Apply functional extensionality with
x1 x2,
Descr_iii x0 x2.
Let x3 of type ι be given.
Claim L3: ∀ x4 : ι → ι → ι . x0 x4 ⟶ x4 x2 x3 = x1 x2 x3
Let x4 of type ι → ι → ι be given.
Assume H3: x0 x4.
Apply H1 with
x1,
x4,
λ x5 x6 : ι → ι → ι . x4 x2 x3 = x6 x2 x3 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Let x5 of type ι → ι → ο be given.
Assume H4: x5 (x4 x2 x3) (x4 x2 x3).
The subproof is completed by applying H4.
Claim L4:
∀ x4 : ι → ι → ι . x0 x4 ⟶ x4 x2 x3 = Descr_iii x0 x2 x3Apply Eps_i_ax with
λ x4 . ∀ x5 : ι → ι → ι . x0 x5 ⟶ x5 x2 x3 = x4,
x1 x2 x3.
The subproof is completed by applying L3.
Apply L4 with
x1.
The subproof is completed by applying H2.
Apply L3 with
λ x2 x3 : ι → ι → ι . x0 x2.
The subproof is completed by applying H2.