Let x0 of type ι be given.
Let x1 of type ι → ι → ο be given.
Assume H0: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2.
Let x2 of type ι be given.
Assume H1: x2 ∈ x0.
Let x3 of type ι be given.
Assume H2: x3 ∈ x0.
Let x4 of type ι be given.
Assume H3: x4 ∈ x0.
Let x5 of type ι be given.
Assume H4: x5 ∈ x0.
Let x6 of type ι be given.
Assume H5: x6 ∈ x0.
Assume H6:
5a3b5.. x1 x2 x3 x4 x5 x6.
Let x7 of type ο be given.
Assume H7:
2f869.. x1 x2 x6 x4 x5 ⟶ (x2 = x3 ⟶ ∀ x8 : ο . x8) ⟶ (x6 = x3 ⟶ ∀ x8 : ο . x8) ⟶ (x4 = x3 ⟶ ∀ x8 : ο . x8) ⟶ (x5 = x3 ⟶ ∀ x8 : ο . x8) ⟶ not (x1 x2 x3) ⟶ x1 x6 x3 ⟶ not (x1 x4 x3) ⟶ not (x1 x5 x3) ⟶ x7.
Apply H6 with
x7.
Apply H9 with
(x2 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x3 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x4 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x5 = x6 ⟶ ∀ x8 : ο . x8) ⟶ not (x1 x2 x6) ⟶ x1 x3 x6 ⟶ not (x1 x4 x6) ⟶ not (x1 x5 x6) ⟶ x7.
Assume H10: x2 = x3 ⟶ ∀ x8 : ο . x8.
Assume H11: x2 = x4 ⟶ ∀ x8 : ο . x8.
Assume H12: x3 = x4 ⟶ ∀ x8 : ο . x8.
Assume H13: x2 = x5 ⟶ ∀ x8 : ο . x8.
Assume H14: x3 = x5 ⟶ ∀ x8 : ο . x8.
Assume H15: x4 = x5 ⟶ ∀ x8 : ο . x8.
Assume H16:
not (x1 x2 x3).
Assume H17:
not (x1 x2 x4).
Assume H18:
not (x1 x3 x4).
Assume H19:
not (x1 x2 x5).
Assume H20:
not (x1 x3 x5).
Assume H21: x1 x4 x5.
Assume H22: x2 = x6 ⟶ ∀ x8 : ο . x8.
Assume H23: x3 = x6 ⟶ ∀ x8 : ο . x8.
Assume H24: x4 = x6 ⟶ ∀ x8 : ο . x8.
Assume H25: x5 = x6 ⟶ ∀ x8 : ο . x8.
Assume H26:
not (x1 x2 x6).
Assume H27: x1 x3 x6.
Assume H28:
not (x1 x4 x6).
Assume H29:
not (x1 x5 x6).
Let x8 of type ο be given.
Assume H30:
... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ not (x1 x2 ...) ⟶ not (x1 x6 x5) ⟶ x1 x4 x5 ⟶ x8.
Apply H7 leaving 9 subgoals.
The subproof is completed by applying L30.
The subproof is completed by applying H10.
Apply neq_i_sym with
x3,
x6.
The subproof is completed by applying H23.
Apply neq_i_sym with
x3,
x4.
The subproof is completed by applying H12.
Apply neq_i_sym with
x3,
x5.
The subproof is completed by applying H14.
The subproof is completed by applying H16.
Apply H0 with
x3,
x6 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
The subproof is completed by applying H27.
Apply L8 with
x3,
x4 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H18.
Apply L8 with
x3,
x5 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
The subproof is completed by applying H20.