Let x0 of type ι → (ι → ι) → (ι → ι) → (ι → ο) → ο be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Let x3 of type ι → ι be given.
Let x4 of type ι → ο be given.
Assume H0:
∀ x5 : ι → ι . (∀ x6 . prim1 x6 x1 ⟶ x2 x6 = x5 x6) ⟶ ∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4.
Apply unknownprop_de92d664051bae83454b194e690fca03a8272522f5ad5d042a8d91459880662d with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x5 (f482f.. (f482f.. (7ba51.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (f482f.. (f482f.. (7ba51.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (7ba51.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4.
Apply H0 with
f482f.. (f482f.. (7ba51.. x1 x2 x3 x4) (4ae4a.. 4a7ef..)),
f482f.. (f482f.. (7ba51.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))),
decode_p (f482f.. (7ba51.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) leaving 3 subgoals.
The subproof is completed by applying unknownprop_872a556b6150b80f45b2d59f5ba6ad3e271d91f36748d2151b2959da9e1f2fee with x1, x2, x3, x4.
The subproof is completed by applying unknownprop_74e84643bb95b0b603e98176f2e9987ba527acafe21e0b3b2868c9969f37eb5b with x1, x2, x3, x4.
Let x5 of type ι be given.
Apply unknownprop_2ff6fecd0198d6fdbd7de25cb9eef11591da871697969a2154ebf8ef906f88c2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x4 x5) x6 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x4 x5.