Let x0 of type ι → (ι → ι → ι) → (ι → ι → ι) → (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 → ι 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 → 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_8ebec66f6d64500c28415cb47e1dcef5c09400d8f1226f5f88fa47a37c141c6e with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x6 (e3162.. (f482f.. (1e782.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (e3162.. (f482f.. (1e782.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (1e782.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. (1e782.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4 x5.
Apply unknownprop_e165443d2d93f0c47db1283918ab4f68381488f58b3de314ab05966c6edc3a6b with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x1 (e3162.. (f482f.. (1e782.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (e3162.. (f482f.. (1e782.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (1e782.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 = x0 x1 x2 x3 x4 x5.
Apply H0 with
e3162.. (f482f.. (1e782.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..)),
e3162.. (f482f.. (1e782.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..))),
decode_p (f482f.. (1e782.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) leaving 3 subgoals.
The subproof is completed by applying unknownprop_056d2098521d1effe203af579b43a9f58ec6ecbc89c2db56b14d5f51c4782b11 with x1, x2, x3, x4, x5.
The subproof is completed by applying unknownprop_c207b2ee22cab49a0eb83d0c8bdd542ccf4264d5fb58ae540c35dd1923a2ab8a with x1, x2, x3, x4, x5.
Let x6 of type ι be given.
Apply unknownprop_6bfdeef306f555a0cd017e3b1a5b615444a3d5009b18ad9e9c1cb7b5264be46e 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.