Let x0 of type ι → (ι → ι → ι) → (ι → ι) → (ι → ο) → ι be given.
Let x1 of type ι be given.
Let x2 of type ι → 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 → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ 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_671251b916faa9f30bde4669a014519d7c43936df2d92d60762c27df7b65a3f5 with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x5 (e3162.. (f482f.. (1670d.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (f482f.. (f482f.. (1670d.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (1670d.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4.
Apply H0 with
e3162.. (f482f.. (1670d.. x1 x2 x3 x4) (4ae4a.. 4a7ef..)),
f482f.. (f482f.. (1670d.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))),
decode_p (f482f.. (1670d.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) leaving 3 subgoals.
The subproof is completed by applying unknownprop_201c00f4dc546c1c361481e6df2db6a0eaaed987dfc99dd72aafe0c44e14d85c with x1, x2, x3, x4.
The subproof is completed by applying unknownprop_e22c9b8bbaf8c317cab2c404e4bfbf3bd3c6d0de5b90be91aa49bf9920fa5b22 with x1, x2, x3, x4.
Let x5 of type ι be given.
Apply unknownprop_ec7e3e177a83a4e42d38d8856b0fff216c61fbfa7dac384f7cc308a89915b411 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.