Let x0 of type ι → ((ι → ο) → ο) → (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 → ο be given.
Let x4 of type ι be given.
Assume H0:
∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ο . (∀ x7 . prim1 x7 x1 ⟶ iff (x3 x7) (x6 x7)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4.
Apply unknownprop_b585ddd2d1f87e91de57052e16fa9fc9f7f1c8519d74bb4e06810b6ca053a11d with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x5 (decode_c (f482f.. (c42e1.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (decode_p (f482f.. (c42e1.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (c42e1.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) = x0 x1 x2 x3 x4.
Apply unknownprop_469b03203711fed4c87fd71fa2d6c39cf5b64ad461f7d8704fa32e6ec790e785 with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x1 (decode_c (f482f.. (c42e1.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (decode_p (f482f.. (c42e1.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) x5 = x0 x1 x2 x3 x4.
Apply H0 with
decode_c (f482f.. (c42e1.. x1 x2 x3 x4) (4ae4a.. 4a7ef..)),
decode_p (f482f.. (c42e1.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) leaving 2 subgoals.
Let x5 of type ι → ο be given.
Assume H1:
∀ x6 . x5 x6 ⟶ prim1 x6 x1.
Apply unknownprop_683bfcbe4bb37f8c2f4fa117c5308204038b13f46c0af4ff7943cad8646fcf56 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x2 x5) x6 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x2 x5.
Let x5 of type ι be given.
Apply unknownprop_2eb06be09378db168b599d1160cfb8b63718d4184d83ea92ee29734845bee9b8 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x3 x5) x6 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x3 x5.