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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: 76607.. x0.
Assume H1: 76607.. x1.
Assume H2: ∀ x2 x3 . iff (prim1 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x4 . If_i (x4 = 4a7ef..) x2 x3)) x0) (prim1 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x4 . If_i (x4 = 4a7ef..) x2 x3)) x1).
Apply set_ext with x0, x1 leaving 2 subgoals.
Let x2 of type ι be given.
Assume H3: prim1 x2 x0.
Apply H0 with x2, prim1 x2 x1 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H4: (λ x4 . ∃ x5 . x2 = 0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x6 . If_i (x6 = 4a7ef..) x4 x5)) x3.
Apply H4 with prim1 x2 x1.
Let x4 of type ι be given.
Assume H5: x2 = 0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x5 . If_i (x5 = 4a7ef..) x3 x4).
Apply H5 with λ x5 x6 . prim1 x6 x1.
Apply H2 with x3, x4, prim1 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x5 . If_i (x5 = 4a7ef..) x3 x4)) x1.
Assume H6: prim1 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x5 . If_i (x5 = 4a7ef..) x3 x4)) x0 ⟶ prim1 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x5 . If_i (x5 = 4a7ef..) x3 x4)) x1.
Assume H7: prim1 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x5 . If_i (x5 = 4a7ef..) x3 x4)) x1 ⟶ prim1 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x5 . If_i (x5 = 4a7ef..) x3 x4)) x0.
Apply H6.
Apply H5 with λ x5 x6 . prim1 x5 x0.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Assume H3: prim1 x2 x1.
Apply H1 with x2, prim1 x2 x0 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H4: (λ x4 . ∃ x5 . x2 = 0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x6 . If_i (x6 = 4a7ef..) x4 x5)) x3.
Apply H4 with prim1 x2 x0.
Let x4 of type ι be given.
Assume H5: x2 = 0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x5 . If_i (x5 = 4a7ef..) x3 x4).
Apply H5 with λ x5 x6 . prim1 x6 x0.
Apply H2 with x3, x4, prim1 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x5 . If_i (x5 = 4a7ef..) x3 x4)) x0.
Assume H6: prim1 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x5 . If_i (x5 = 4a7ef..) x3 x4)) x0 ⟶ prim1 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x5 . If_i (x5 = 4a7ef..) x3 x4)) x1.
Assume H7: prim1 (0fc90.. ... ...) ... ⟶ prim1 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x5 . If_i (x5 = 4a7ef..) x3 x4)) x0.
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