Let x0 of type ι → (ι → ο) → (ι → ο) → ο be given.
Let x1 of type ι be given.
Let x2 of type ι → ο be given.
Let x3 of type ι → ο be given.
Assume H0:
∀ x4 : ι → ο . (∀ x5 . prim1 x5 x1 ⟶ iff (x2 x5) (x4 x5)) ⟶ ∀ x5 : ι → ο . (∀ x6 . prim1 x6 x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3.
Apply unknownprop_5a3da60961dc828c3b054c11b04e3fdd5c1744838c2dc7756fea2affd3e32f1d with
x1,
x2,
x3,
λ x4 x5 . x0 x4 (decode_p (f482f.. (2c81e.. x1 x2 x3) (4ae4a.. 4a7ef..))) (decode_p (f482f.. (2c81e.. x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..)))) = x0 x1 x2 x3.
Apply H0 with
decode_p (f482f.. (2c81e.. x1 x2 x3) (4ae4a.. 4a7ef..)),
decode_p (f482f.. (2c81e.. x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) leaving 2 subgoals.
Let x4 of type ι be given.
Apply unknownprop_f89b2eb83b1375e36cd0ad303ce6605f299e4461f16d1e3558e0856c6ba04a59 with
x1,
x2,
x3,
x4,
λ x5 x6 : ο . iff (x2 x4) x5 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x2 x4.
Let x4 of type ι be given.
Apply unknownprop_353e1ebd952cdfa03484f2eb926c90f10068fbb8a5e3cf11a3a5f72589e7c77c with
x1,
x2,
x3,
x4,
λ x5 x6 : ο . iff (x3 x4) x5 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x3 x4.