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.

Assume H0: ∀ x3 : ι → ο . (∀ x4 . x3 x4 ⟶ prim1 x4 x0) ⟶ iff (x1 x3) (x2 x3).

Claim L1: e0e40.. x0 x1 = e0e40.. x0 x2

Apply unknownprop_35ee954b0de81ace4d484d57278ef6dea3fd2cb486e752fb5784dfc8cd9b7c4a with x0, x1, x2.

The subproof is completed by applying H0.

Apply L1 with λ x3 x4 . 0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x5 . If_i (x5 = 4a7ef..) x0 (e0e40.. x0 x1)) = 0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x5 . If_i (x5 = 4a7ef..) x0 x3).

Let x3 of type ι → ι → ο be given.

Assume H2: x3 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x4 . If_i (x4 = 4a7ef..) x0 (e0e40.. x0 x1))) (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x4 . If_i (x4 = 4a7ef..) x0 (e0e40.. x0 x1))).

The subproof is completed by applying H2.

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Proofgold Proof