Let x0 of type ι be given.
Let x1 of type ι → ι → ο be given.
Assume H0: ∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2.
Let x2 of type ι be given.
Assume H1: x2 ∈ x0.
Let x3 of type ι be given.
Apply SepE with
x0,
λ x4 . and (x2 = x4 ⟶ ∀ x5 : ο . x5) (x1 x2 x4),
x3,
x2 ∈ DirGraphOutNeighbors x0 x1 x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H3: x3 ∈ x0.
Assume H4:
and (x2 = x3 ⟶ ∀ x4 : ο . x4) (x1 x2 x3).
Apply H4 with
x2 ∈ DirGraphOutNeighbors x0 x1 x3.
Assume H5: x2 = x3 ⟶ ∀ x4 : ο . x4.
Assume H6: x1 x2 x3.
Apply SepI with
x0,
λ x4 . and (x3 = x4 ⟶ ∀ x5 : ο . x5) (x1 x3 x4),
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply andI with
x3 = x2 ⟶ ∀ x4 : ο . x4,
x1 x3 x2 leaving 2 subgoals.
Apply neq_i_sym with
x2,
x3.
The subproof is completed by applying H5.
Apply H0 with
x2,
x3.
The subproof is completed by applying H6.