Let x0 of type ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → ο 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.
Assume H0:
∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4.
Apply unknownprop_36c42de3fa64ba0d931d2bce5c95c24271e7ca12e15351b214ce939787147f70 with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x5 (e3162.. (f482f.. (8f64a.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (e3162.. (f482f.. (8f64a.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (8f64a.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4.
Apply H0 with
e3162.. (f482f.. (8f64a.. x1 x2 x3 x4) (4ae4a.. 4a7ef..)),
e3162.. (f482f.. (8f64a.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))),
decode_p (f482f.. (8f64a.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) leaving 3 subgoals.
The subproof is completed by applying unknownprop_4e842e2c30ccf0c3564e02682ecde09675764a8b5663da987f76b91e34225304 with x1, x2, x3, x4.
The subproof is completed by applying unknownprop_521087284df84a924fcf4bcd5859c5772f0f0c1ecfc3853224d5653fe7e0ef2c with x1, x2, x3, x4.
Let x5 of type ι be given.
Apply unknownprop_cd131d10c369b276d5ab02bbaa6503cd63c1cdda813ea32dfc1d611915267593 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x4 x5) x6 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x4 x5.