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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: set_of_pairs x0.
Assume H1: set_of_pairs x1.
Assume H2: ∀ x2 x3 . iff (lam 2 (λ x4 . If_i (x4 = 0) x2 x3)x0) (lam 2 (λ x4 . If_i (x4 = 0) x2 x3)x1).
Apply set_ext with x0, x1 leaving 2 subgoals.
Let x2 of type ι be given.
Assume H3: x2x0.
Apply H0 with x2, x2x1 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H4: (λ x4 . ∃ x5 . x2 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)) x3.
Apply H4 with x2x1.
Let x4 of type ι be given.
Assume H5: x2 = lam 2 (λ x5 . If_i (x5 = 0) x3 x4).
Apply H5 with λ x5 x6 . x6x1.
Apply H2 with x3, x4, lam 2 (λ x5 . If_i (x5 = 0) x3 x4)x1.
Assume H6: lam 2 (λ x5 . If_i (x5 = 0) x3 x4)x0lam 2 (λ x5 . If_i (x5 = 0) x3 x4)x1.
Assume H7: lam 2 (λ x5 . If_i (x5 = 0) x3 x4)x1lam 2 (λ x5 . If_i (x5 = 0) x3 x4)x0.
Apply H6.
Apply H5 with λ x5 x6 . x5x0.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Assume H3: x2x1.
Apply H1 with x2, x2x0 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H4: (λ x4 . ∃ x5 . x2 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)) x3.
Apply H4 with x2x0.
Let x4 of type ι be given.
Assume H5: x2 = lam 2 (λ x5 . If_i (x5 = 0) x3 x4).
Apply H5 with λ x5 x6 . x6x0.
Apply H2 with x3, x4, lam 2 (λ x5 . If_i (x5 = 0) x3 x4)x0.
Assume H6: lam 2 (λ x5 . If_i (x5 = 0) x3 x4)x0lam 2 (λ x5 . If_i (x5 = 0) x3 x4)x1.
Assume H7: lam 2 (λ x5 . If_i (x5 = 0) x3 x4)x1lam 2 (λ x5 . If_i (x5 = 0) x3 x4)x0.
Apply H7.
Apply H5 with λ x5 x6 . x5x1.
The subproof is completed by applying H3.