Let x0 of type ι → (ι → ι → ι) → (ι → ι → ο) → ι → ι → ο be given.
Let x1 of type ι be given.
Let x2 of type ι → 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 → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ 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_4d3b1552f811436b8f29af8fa2db6a6f0ed851417fe4ea754fadac215fbbceef with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x6 (e3162.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4 x5.
Apply unknownprop_ddb82a3c52c030568b927ce40224fa285ba790e62b2413b7fca117d1b06927bc with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x1 (e3162.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4 x5.
Apply unknownprop_a2784db0fbde3c5924110ee44b88db7f7be64c189a40d459e45f4debf5b5aa45 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x1 (e3162.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) x4 x6 = x0 x1 x2 x3 x4 x5.
Apply H0 with
e3162.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..)),
2b2e3.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..))) leaving 2 subgoals.
The subproof is completed by applying unknownprop_1577ea96652eddec718222b40e2ddf7792be06e6a335faa6da4659776d8b2b7f with x1, x2, x3, x4, x5.
Let x6 of type ι be given.
Let x7 of type ι be given.
Apply unknownprop_303a5dcd96ccd0795a0c1c2f0ababef222f2dd40fc6be62386d34762cfdb65c5 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.