Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι be given.
Assume H0:
x2 ∈ lam x0 (λ x3 . x1 x3).
Let x3 of type ο be given.
Assume H1:
∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x1 x4 ⟶ x2 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5) ⟶ x3.
Apply lamE2 with
x0,
x1,
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x4 of type ι be given.
Assume H2:
(λ x5 . and (x5 ∈ x0) (∃ x6 . and (x6 ∈ x1 x5) (x2 = lam 2 (λ x7 . If_i (x7 = 0) x5 x6)))) x4.
Apply H2 with
x3.
Assume H3: x4 ∈ x0.
Assume H4:
∃ x5 . and (x5 ∈ x1 x4) (x2 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)).
Apply H4 with
x3.
Let x5 of type ι be given.
Assume H5:
(λ x6 . and (x6 ∈ x1 x4) (x2 = lam 2 (λ x7 . If_i (x7 = 0) x4 x6))) x5.
Apply H5 with
x3.
Assume H6: x5 ∈ x1 x4.
Assume H7:
x2 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5).
Apply H1 with
x4,
x5 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H6.
The subproof is completed by applying H7.