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.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Let x8 of type ι be given.
Let x9 of type ι be given.
Apply beta with
10,
λ x10 . If_i (x10 = 0) x0 (If_i (x10 = 1) x1 (If_i (x10 = 2) x2 (If_i (x10 = 3) x3 (If_i (x10 = 4) x4 (If_i (x10 = 5) x5 (If_i (x10 = 6) x6 (If_i (x10 = 7) x7 (If_i (x10 = 8) x8 x9)))))))),
0,
λ x10 x11 . x11 = x0 leaving 2 subgoals.
The subproof is completed by applying unknownprop_d949f57c5dfaf614a6f4f3766686f51d9eb716e4d0cf9f16865af268fc065aeb.
Apply If_i_1 with
0 = 0,
x0,
If_i (0 = 1) x1 (If_i (0 = 2) x2 (If_i (0 = 3) x3 (If_i (0 = 4) x4 (If_i (0 = 5) x5 (If_i (0 = 6) x6 (If_i (0 = 7) x7 (If_i (0 = 8) x8 x9))))))).
Let x10 of type ι → ι → ο be given.
Assume H0: x10 0 0.
The subproof is completed by applying H0.