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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ιι be given.
Let x2 of type ιιο be given.
Let x3 of type ι be given.
Assume H0: x3Sep2 x0 x1 x2.
Apply Sep2E with x0, x1, x2, x3, ∃ x4 x5 . x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x4 of type ι be given.
Assume H1: (λ x5 . and (x5x0) (∃ x6 . and (x6x1 x5) (and (x3 = lam 2 (λ x7 . If_i (x7 = 0) x5 x6)) (x2 x5 x6)))) x4.
Apply H1 with ∃ x5 x6 . x3 = lam 2 (λ x7 . If_i (x7 = 0) x5 x6).
Assume H2: x4x0.
Assume H3: ∃ x5 . and (x5x1 x4) (and (x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)) (x2 x4 x5)).
Apply H3 with ∃ x5 x6 . x3 = lam 2 (λ x7 . If_i (x7 = 0) x5 x6).
Let x5 of type ι be given.
Assume H4: (λ x6 . and (x6x1 x4) (and (x3 = lam 2 (λ x7 . If_i (x7 = 0) x4 x6)) (x2 x4 x6))) x5.
Apply H4 with ∃ x6 x7 . x3 = lam 2 (λ x8 . If_i (x8 = 0) x6 x7).
Assume H5: x5x1 x4.
Assume H6: and (x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)) (x2 x4 x5).
Apply H6 with ∃ x6 x7 . x3 = lam 2 (λ x8 . If_i (x8 = 0) x6 x7).
Assume H7: x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5).
Assume H8: x2 x4 x5.
Let x6 of type ο be given.
Assume H9: ∀ x7 . (∃ x8 . x3 = lam 2 (λ x9 . If_i (x9 = 0) x7 x8))x6.
Apply H9 with x4.
Let x7 of type ο be given.
Assume H10: ∀ x8 . x3 = lam 2 (λ x9 . If_i (x9 = 0) x4 x8)x7.
Apply H10 with x5.
The subproof is completed by applying H7.