Let x0 of type ο be given.
Apply prop_ext_2 with
6fb7f.. (λ x1 x2 . If_i x0 x2 x1),
x0 leaving 2 subgoals.
Apply dneg with
x0.
Apply neq_0_1.
set y1 to be 0
set y2 to be 1
Claim L2: ∀ x3 : ι → ο . x3 y2 ⟶ x3 y1
Let x3 of type ι → ο be given.
Assume H2: x3 1.
set y4 to be λ x4 . x3
Apply H0 with
λ x5 x6 : ι → ι → ι . x5 1 0 = 1,
λ x5 . y4 leaving 2 subgoals.
Apply If_i_0 with
x3,
0,
1.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying L3.
Let x3 of type ι → ι → ο be given.
Apply L2 with
λ x4 . x3 x4 y2 ⟶ x3 y2 x4.
Assume H3: x3 y2 y2.
The subproof is completed by applying H3.
Assume H0: x0.
Apply functional extensionality with
λ x1 x2 . If_i x0 x2 x1,
ChurchBoolFal.
Let x1 of type ι be given.
Apply functional extensionality with
λ x2 . If_i x0 x2 x1,
λ x2 . x2.
Let x2 of type ι be given.
Apply If_i_1 with
x0,
x2,
x1.
The subproof is completed by applying H0.