Let x0 of type ι → (ι → ι → ι) → (ι → ι → ο) → (ι → ο) → ι → ι 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 → ο 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)) ⟶ ∀ 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_ad9008b6f2810a0e7af71389371f7ebb2072060b97071ea08eac6648b1392012 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x6 (e3162.. (f482f.. (02b3f.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (02b3f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (02b3f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. (02b3f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4 x5.
Apply unknownprop_cbf73d6429ec8fb960c667c86ebf36a6e504cfb29f13f37b32891904f10afd24 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x1 (e3162.. (f482f.. (02b3f.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (02b3f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (02b3f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 = x0 x1 x2 x3 x4 x5.
Apply H0 with
e3162.. (f482f.. (02b3f.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..)),
2b2e3.. (f482f.. (02b3f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..))),
decode_p (f482f.. (02b3f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) leaving 3 subgoals.
The subproof is completed by applying unknownprop_68a0f865e52b1f2086cf38474b5b6a79876dd054c0b1d3fd70ebad35e03e9edb with x1, x2, x3, x4, x5.
Let x6 of type ι be given.
Let x7 of type ι be given.
Apply unknownprop_903218825c8191e8fbd41c581818c0ede6ad6eef9ed65f1a49cd074f0c204395 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.
Let x6 of type ι be given.
Apply unknownprop_cfa405b88b73d28820f49d9bf44e58432ec35b7af75bdb9faca7a7821189bcd7 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.