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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type (ι → ο) → ο be given.
Let x2 of type (ι → ο) → ο be given.
Let x3 of type ι → ι → ο be given.
Let x4 of type ι → ι → ο be given.
Assume H0: ∀ x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x0) ⟶ iff (x1 x5) (x2 x5).
Assume H1: ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ iff (x3 x5 x6) (x4 x5 x6).
Claim L2: e0e40.. x0 x1 = e0e40.. x0 x2
Apply unknownprop_35ee954b0de81ace4d484d57278ef6dea3fd2cb486e752fb5784dfc8cd9b7c4a with x0, x1, x2.
The subproof is completed by applying H0.
Apply L2 with λ x5 x6 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x7 . If_i (x7 = 4a7ef..) x0 (If_i (x7 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (d2155.. x0 x3))) = 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x7 . If_i (x7 = 4a7ef..) x0 (If_i (x7 = 4ae4a.. 4a7ef..) x5 (d2155.. x0 x4))).
Claim L3: d2155.. x0 x3 = d2155.. x0 x4
Apply unknownprop_75d5b46497f20dc30e2e5351a60197c4fa9d445bc23c6c8245597bb858180907 with x0, x3, x4.
The subproof is completed by applying H1.
Apply L3 with λ x5 x6 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x7 . If_i (x7 = 4a7ef..) x0 (If_i (x7 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (d2155.. x0 x3))) = 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x7 . If_i (x7 = 4a7ef..) x0 (If_i (x7 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) x5)).
Let x5 of type ι → ι → ο be given.
Assume H4: x5 (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (d2155.. x0 x3)))) (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (d2155.. x0 x3)))).
The subproof is completed by applying H4.
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