Let x0 of type ι → (ι → ι) → (ι → ι → ο) → ι → ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Let x3 of type ι → ι → ο be given.
Let x4 of type ι be given.
Assume H0:
∀ x5 : ι → ι . (∀ x6 . prim1 x6 x1 ⟶ x2 x6 = x5 x6) ⟶ ∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (x3 x7 x8) (x6 x7 x8)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4.
Apply unknownprop_0cc92ec4e0e6dab19c106c8af8774f1cda73073bc0b40c4ccb1840d7af9b0b0d with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x5 (f482f.. (f482f.. (1bcc7.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (1bcc7.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (1bcc7.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) = x0 x1 x2 x3 x4.
Apply unknownprop_e736d87fb1c82f1653455bc78e80c3d1deb331bc1d61d105d9782506c903a8fd with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x1 (f482f.. (f482f.. (1bcc7.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (1bcc7.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) x5 = x0 x1 x2 x3 x4.
Apply H0 with
f482f.. (f482f.. (1bcc7.. x1 x2 x3 x4) (4ae4a.. 4a7ef..)),
2b2e3.. (f482f.. (1bcc7.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) leaving 2 subgoals.
The subproof is completed by applying unknownprop_0fd914315b038e6cc084f062f01503a01c3efa5b54d8e88f9d1216b20fcbcdc3 with x1, x2, x3, x4.
Let x5 of type ι be given.
Let x6 of type ι be given.
Apply unknownprop_022211655377b981fe7e19d29dd3c365db553d8685042e45a1cf4cc105d846fc with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x3 x5 x6) x7 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 x5 x6.