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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 ιιο be given.
Assume H0: ∀ x4 . x4x0∀ x5 . x5x1 x4iff (x2 x4 x5) (x3 x4 x5).
Apply set_of_pairs_ext with Sep2 x0 x1 x2, Sep2 x0 x1 x3 leaving 3 subgoals.
The subproof is completed by applying Sep2_set_of_pairs with x0, x1, x2.
The subproof is completed by applying Sep2_set_of_pairs with x0, x1, x3.
Let x4 of type ι be given.
Let x5 of type ι be given.
Apply iffI with lam 2 (λ x6 . If_i (x6 = 0) x4 x5)Sep2 x0 x1 x2, lam 2 (λ x6 . If_i (x6 = 0) x4 x5)Sep2 x0 x1 x3 leaving 2 subgoals.
Assume H1: lam 2 (λ x6 . If_i (x6 = 0) x4 x5)Sep2 x0 x1 x2.
Apply Sep2E' with x0, x1, x2, x4, x5, lam 2 (λ x6 . If_i (x6 = 0) x4 x5)Sep2 x0 x1 x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H2: and (x4x0) (x5x1 x4).
Assume H3: x2 x4 x5.
Apply H2 with lam 2 (λ x6 . If_i (x6 = 0) x4 x5)Sep2 x0 x1 x3.
Assume H4: x4x0.
Assume H5: x5x1 x4.
Apply Sep2I with x0, x1, x3, x4, x5 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply H0 with x4, x5, x3 x4 x5 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Assume H6: x2 x4 x5x3 x4 x5.
Assume H7: x3 x4 x5x2 x4 x5.
Apply H6.
The subproof is completed by applying H3.
Assume H1: lam 2 (λ x6 . If_i (x6 = 0) x4 x5)Sep2 x0 x1 x3.
Apply Sep2E' with x0, x1, x3, x4, x5, lam 2 (λ x6 . If_i (x6 = 0) x4 x5)Sep2 x0 x1 x2 leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H2: and (x4x0) (x5x1 x4).
Assume H3: x3 x4 x5.
Apply H2 with lam 2 (λ x6 . If_i (x6 = 0) x4 x5)Sep2 x0 x1 x2.
Assume H4: x4x0.
Assume H5: x5x1 x4.
Apply Sep2I with x0, x1, x2, x4, x5 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply H0 with x4, x5, x2 x4 x5 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Assume H6: x2 x4 x5x3 x4 x5.
Assume H7: x3 x4 x5x2 x4 x5.
Apply H7.
The subproof is completed by applying H3.