Let x0 of type ι → (ι → ι → ι) → (ι → ι → ο) → 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.
Assume H0:
∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = x4 x5 x6) ⟶ ∀ 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_2e4ad16a724aa9ecb1f2d0714afea104f57cf750321b796a86385476dc14b16d with
x1,
x2,
x3,
λ x4 x5 . x0 x4 (e3162.. (f482f.. (b5cc3.. x1 x2 x3) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (b5cc3.. x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..)))) = x0 x1 x2 x3.
Apply H0 with
e3162.. (f482f.. (b5cc3.. x1 x2 x3) (4ae4a.. 4a7ef..)),
2b2e3.. (f482f.. (b5cc3.. x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) leaving 2 subgoals.
The subproof is completed by applying unknownprop_15cd99d9ca461ad2237964118d360f2606f8cb1d2c65d040d581982ac836b5ae with x1, x2, x3.
Let x4 of type ι be given.
Let x5 of type ι be given.
Apply unknownprop_1b1870aad03d524430ecdbf45bff93c05749be68abe1a80eafa30ff7d1c8a1e5 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.