Let x0 of type ι → (ι → ι → ι) → (ι → ο) → ι → ο be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ι be given.
Let x3 of type ι → ο be given.
Let x4 of type ι be given.
Assume H0:
∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ο . (∀ x7 . prim1 x7 x1 ⟶ iff (x3 x7) (x6 x7)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4.
Apply unknownprop_848a12d392e8404e9fc3dd49be94ed27ceae23f19931a48e339b5aa92ae6295c with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x5 (e3162.. (f482f.. (60b2c.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (decode_p (f482f.. (60b2c.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (60b2c.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) = x0 x1 x2 x3 x4.
Apply unknownprop_cdda72973dc0b5b6f0b46928fa4486ae7f28c61d3aa53ff1cbdff5c714bc6b4a with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x1 (e3162.. (f482f.. (60b2c.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (decode_p (f482f.. (60b2c.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) x5 = x0 x1 x2 x3 x4.
Apply H0 with
e3162.. (f482f.. (60b2c.. x1 x2 x3 x4) (4ae4a.. 4a7ef..)),
decode_p (f482f.. (60b2c.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) leaving 2 subgoals.
The subproof is completed by applying unknownprop_83df285a23471f1da5b54cf57ce9aa7010b1ec60eb9a7332adfdd0775116fcef with x1, x2, x3, x4.
Let x5 of type ι be given.
Apply unknownprop_92ebd3f7147f6de662aed94671adec8d6b4e8e50d19f8fe5ae090016dceb1248 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x3 x5) x6 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x3 x5.