Let x0 of type ι → (ι → ι) → (ι → ι → ο) → ο 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.
Assume H0:
∀ x4 : ι → ι . (∀ x5 . prim1 x5 x1 ⟶ x2 x5 = x4 x5) ⟶ ∀ x5 : ι → ι → ο . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ iff (x3 x6 x7) (x5 x6 x7)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3.
Apply unknownprop_56fff63ae4aaffa24f98b7fb5e034afaf3bd48ea6cf337512ab94a8fb832e90d with
x1,
x2,
x3,
λ x4 x5 . x0 x4 (f482f.. (f482f.. (73737.. x1 x2 x3) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (73737.. x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..)))) = x0 x1 x2 x3.
Apply H0 with
f482f.. (f482f.. (73737.. x1 x2 x3) (4ae4a.. 4a7ef..)),
2b2e3.. (f482f.. (73737.. x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) leaving 2 subgoals.
The subproof is completed by applying unknownprop_0eba4a1e8a6eb85bd93458d64288a202dddf6558775b796d073829b1ed4aed10 with x1, x2, x3.
Let x4 of type ι be given.
Let x5 of type ι be given.
Apply unknownprop_d3745c59fd830feec1cf172211f6f2a659a425d7b1b42063659cffa5062c6422 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x3 x4 x5) x6 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 x4 x5.