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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ιιο be given.
Assume H0: ∀ x2 x3 . x1 x2 x3x1 x3 x2.
Let x2 of type ι be given.
Assume H1: x2x0.
Let x3 of type ι be given.
Assume H2: x3DirGraphOutNeighbors x0 x1 x2.
Apply SepE with x0, λ x4 . and (x2 = x4∀ x5 : ο . x5) (x1 x2 x4), x3, x2DirGraphOutNeighbors x0 x1 x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H3: x3x0.
Assume H4: and (x2 = x3∀ x4 : ο . x4) (x1 x2 x3).
Apply H4 with x2DirGraphOutNeighbors 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.