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.
Let x4 of type ι be given.
Apply beta with
5,
λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 (If_i (x5 = 3) x3 x4))),
4,
λ x5 x6 . x6 = x4 leaving 2 subgoals.
The subproof is completed by applying In_4_5.
Apply If_i_0 with
4 = 0,
x0,
If_i (4 = 1) x1 (If_i (4 = 2) x2 (If_i (4 = 3) x3 x4)),
λ x5 x6 . x6 = x4 leaving 2 subgoals.
The subproof is completed by applying neq_4_0.
Apply If_i_0 with
4 = 1,
x1,
If_i (4 = 2) x2 (If_i (4 = 3) x3 x4),
λ x5 x6 . x6 = x4 leaving 2 subgoals.
The subproof is completed by applying neq_4_1.
Apply If_i_0 with
4 = 2,
x2,
If_i (4 = 3) x3 x4,
λ x5 x6 . x6 = x4 leaving 2 subgoals.
The subproof is completed by applying neq_4_2.
Apply If_i_0 with
4 = 3,
x3,
x4.
The subproof is completed by applying neq_4_3.