Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Apply beta with
4,
λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 (If_i (x4 = 2) x2 x3)),
2,
λ x4 x5 . x5 = x2 leaving 2 subgoals.
The subproof is completed by applying In_2_4.
Apply If_i_0 with
2 = 0,
x0,
If_i (2 = 1) x1 (If_i (2 = 2) x2 x3),
λ x4 x5 . x5 = x2 leaving 2 subgoals.
The subproof is completed by applying neq_2_0.
Apply If_i_0 with
2 = 1,
x1,
If_i (2 = 2) x2 x3,
λ x4 x5 . x5 = x2 leaving 2 subgoals.
The subproof is completed by applying neq_2_1.
Apply If_i_1 with
2 = 2,
x2,
x3.
Let x4 of type ι → ι → ο be given.
Assume H0: x4 2 2.
The subproof is completed by applying H0.