Let x0 of type ι → (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 → 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 → 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_42a095fe3b02ba45094f1e14a1ac5ada5d6d439811f9566f16b43b84348ab77f with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x5 (e3162.. (f482f.. (4d5a4.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (4d5a4.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (4d5a4.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) = x0 x1 x2 x3 x4.
Apply unknownprop_c215c3321ba335d997a7897a6163689a168ec4bedb2b0c5d919146b523668141 with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x1 (e3162.. (f482f.. (4d5a4.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (4d5a4.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) x5 = x0 x1 x2 x3 x4.
Apply H0 with
e3162.. (f482f.. (4d5a4.. x1 x2 x3 x4) (4ae4a.. 4a7ef..)),
2b2e3.. (f482f.. (4d5a4.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) leaving 2 subgoals.
The subproof is completed by applying unknownprop_f06b0a67dd5bbbeac58fd338a725d744b2a4efcf8fe27966dfa1185aa34a901c with x1, x2, x3, x4.
Let x5 of type ι be given.
Let x6 of type ι be given.
Apply unknownprop_e57334b898c7a3e2da00fbb9975b5ba2aef7c7a09b0668a091b2dc60dedd57bf 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.