Let x0 of type ι → ((ι → ο) → ο) → (ι → ι → ο) → ι → ι be given.
Let x1 of type ι be given.
Let x2 of type (ι → ο) → ο be given.
Let x3 of type ι → ι → ο be given.
Let x4 of type ι be given.
Assume H0:
∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (x3 x7 x8) (x6 x7 x8)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4.
Apply unknownprop_d754eaf2b0fb19fb3d886b32f1b60dbaa8db7ea48022e79931c69cf7717a6d38 with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x5 (decode_c (f482f.. (d8d01.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (d8d01.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (d8d01.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) = x0 x1 x2 x3 x4.
Apply unknownprop_a619e6c9c650f9242f9eb8820c4a42d5f40f543a1d8fbf54862d930cf3bb5b27 with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x1 (decode_c (f482f.. (d8d01.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (d8d01.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) x5 = x0 x1 x2 x3 x4.
Apply H0 with
decode_c (f482f.. (d8d01.. x1 x2 x3 x4) (4ae4a.. 4a7ef..)),
2b2e3.. (f482f.. (d8d01.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) leaving 2 subgoals.
Let x5 of type ι → ο be given.
Assume H1:
∀ x6 . x5 x6 ⟶ prim1 x6 x1.
Apply unknownprop_1c403a64ca50b475d6bc7af7f90763f88ed5ce47216d11d62787136b399d2c54 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x2 x5) x6 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x2 x5.
Let x5 of type ι be given.
Let x6 of type ι be given.
Apply unknownprop_b476b153a6360678b1439cc46f7117cebbfe735588ee20ff35fcd444ab0ae523 with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x3 x5 x6) x7 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying iff_refl with x3 x5 x6.