Let x0 of type ι → (ι → ι) → (ι → ι → ο) → ι → ι → ο be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Let x3 of type ι → ι → ο be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Assume H0:
∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι → ο . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ iff (x3 x8 x9) (x7 x8 x9)) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5.
Apply unknownprop_0e44755d1847d4635082b55b8624b90b6656b4694255a868b99566ebb2ae75f2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x6 (f482f.. (f482f.. (bd517.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (bd517.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (bd517.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (bd517.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4 x5.
Apply unknownprop_c38a1b4a91b8593e922312e9d9c1bc283d51d451b015b37c53e8061dd2066ce6 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x1 (f482f.. (f482f.. (bd517.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (bd517.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (f482f.. (bd517.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4 x5.
Apply unknownprop_022e3275e73cbdf94153df7d27b74c51a653a052083e928a13fefdb2c4e7b8ac with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x1 (f482f.. (f482f.. (bd517.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (bd517.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) x4 x6 = x0 x1 x2 x3 x4 x5.
Apply H0 with
f482f.. (f482f.. (bd517.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..)),
2b2e3.. (f482f.. (bd517.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..))) leaving 2 subgoals.
The subproof is completed by applying unknownprop_eed203900486d4bd074b965d9ca385cbfaa8d93221ce2fd0a72d2075daa3bd7c with x1, x2, x3, x4, x5.
Let x6 of type ι be given.
Let x7 of type ι be given.
Apply unknownprop_c50e43fc18e4f570b032dd6cdd0e8956299d7bdc8e2e8c62771e4644d1ad78a2 with
x1,
x2,
x3,
x4,
x5,
x6,
x7,
λ x8 x9 : ο . iff (x3 x6 x7) x8 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 x6 x7.