Let x0 of type ο be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Apply functional extensionality with
If_iii x0 x1 x2,
x2.
Let x3 of type ι be given.
Apply functional extensionality with
If_iii x0 x1 x2 x3,
x2 x3.
Let x4 of type ι be given.
Apply If_i_0 with
x0,
x1 x3 x4,
x2 x3 x4.
The subproof is completed by applying H0.