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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type (ιο) → ο be given.
Let x2 of type (ιο) → ο be given.
Let x3 of type ιι be given.
Let x4 of type ιι be given.
Let x5 of type ι be given.
Assume H0: ∀ x6 : ι → ο . (∀ x7 . x6 x7prim1 x7 x0)iff (x1 x6) (x2 x6).
Assume H1: ∀ x6 . prim1 x6 x0x3 x6 = x4 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 x6 x7 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x8 . If_i (x8 = 4a7ef..) x0 (If_i (x8 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x8 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x3) x5))) = 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x8 . If_i (x8 = 4a7ef..) x0 (If_i (x8 = 4ae4a.. 4a7ef..) x6 (If_i (x8 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x4) x5))).
Claim L3: 0fc90.. x0 x3 = 0fc90.. x0 x4
Apply unknownprop_075a71193d04eff3936ee7246a228619e3dbe0ea2b9d96d40e9b467470ee4a92 with x0, x3, x4.
The subproof is completed by applying H1.
Apply L3 with λ x6 x7 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x8 . If_i (x8 = 4a7ef..) x0 (If_i (x8 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x8 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x3) x5))) = 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x8 . If_i (x8 = 4a7ef..) x0 (If_i (x8 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x8 = 4ae4a.. (4ae4a.. 4a7ef..)) x6 x5))).
Let x6 of type ιιο be given.
Assume H4: x6 (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x7 . If_i (x7 = 4a7ef..) x0 (If_i (x7 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x7 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x3) x5)))) (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x7 . If_i (x7 = 4a7ef..) x0 (If_i (x7 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x7 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x3) x5)))).
The subproof is completed by applying H4.