Let x0 of type ι → ((ι → ο) → ο) → (ι → ι → ι) → (ι → ο) → ι → ο be given.
Let x1 of type ι be given.
Let x2 of type (ι → ο) → ο be given.
Let x3 of type ι → ι → ι be given.
Let x4 of type ι → ο be given.
Let x5 of type ι be given.
Assume H0:
∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5.
Apply unknownprop_65cce1600a86bdb4fcc47c3f21cd540289b2d8a67dba9d2cfdb16c7b60cd3592 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x6 (decode_c (f482f.. (bebf6.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (e3162.. (f482f.. (bebf6.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (bebf6.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. (bebf6.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4 x5.
Apply unknownprop_d541830b9bb76016fdfd9d4a39b99cfc03581aed7733587f376a6613a3d27e18 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x1 (decode_c (f482f.. (bebf6.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (e3162.. (f482f.. (bebf6.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (bebf6.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 = x0 x1 x2 x3 x4 x5.
Apply H0 with
decode_c (f482f.. (bebf6.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..)),
e3162.. (f482f.. (bebf6.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..))),
decode_p (f482f.. (bebf6.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) leaving 3 subgoals.
Let x6 of type ι → ο be given.
Assume H1:
∀ x7 . x6 x7 ⟶ prim1 x7 x1.
Apply unknownprop_893520f3075c4d3a8ba22723b37a813951891a352483f9ee1c79192185721ed8 with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x2 x6) x7 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x2 x6.
The subproof is completed by applying unknownprop_d00271e3409a7f60fd60ced864a095054ad7fb1dd9954a22fb8f485fbf2f84a4 with x1, x2, x3, x4, x5.
Let x6 of type ι be given.
Apply unknownprop_8472672de8e4a6002250f4bca01a91b0df686dd5f362cb55afa19360ce15f05d with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x4 x6) x7 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x4 x6.