Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Assume H0: prim1 x2 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x3 . If_i (x3 = 4a7ef..) x0 x1)).

Claim L1: ∃ x3 . and (prim1 x3 (4ae4a.. (4ae4a.. 4a7ef..))) (∃ x4 . and (prim1 x4 (If_i (x3 = 4a7ef..) x0 x1)) (x2 = aae7a.. x3 x4))

Apply unknownprop_d4dc73f3cbfe4c22363272ac418d035b8e77c49433b0870b96bee8fa9c46bfcf with 4ae4a.. (4ae4a.. 4a7ef..), λ x3 . If_i (x3 = 4a7ef..) x0 x1, x2.

The subproof is completed by applying H0.

Apply exandE_i with λ x3 . prim1 x3 (4ae4a.. (4ae4a.. 4a7ef..)), λ x3 . ∃ x4 . and (prim1 x4 (If_i (x3 = 4a7ef..) x0 x1)) (x2 = aae7a.. x3 x4), ∃ x3 . and (prim1 x3 (4ae4a.. (4ae4a.. 4a7ef..))) (∃ x4 . x2 = aae7a.. x3 x4) leaving 2 subgoals.

The subproof is completed by applying L1.

Let x3 of type ι be given.

Assume H2: prim1 x3 (4ae4a.. (4ae4a.. 4a7ef..)).

Assume H3: ∃ x4 . and (prim1 x4 (If_i (x3 = 4a7ef..) x0 x1)) (x2 = aae7a.. x3 x4).

Apply exandE_i with λ x4 . prim1 x4 (If_i (x3 = 4a7ef..) x0 x1), λ x4 . x2 = aae7a.. x3 x4, ∃ x4 . and (prim1 x4 (4ae4a.. (4ae4a.. 4a7ef..))) (∃ x5 . x2 = aae7a.. x4 x5) leaving 2 subgoals.

The subproof is completed by applying H3.

Let x4 of type ι be given.

Assume H4: prim1 x4 (If_i (x3 = 4a7ef..) x0 x1).

Assume H5: x2 = aae7a.. x3 x4.

Let x5 of type ο be given.

Assume H6: ∀ x6 . and (prim1 x6 (4ae4a.. (4ae4a.. 4a7ef..))) (∃ x7 . x2 = aae7a.. x6 x7) ⟶ x5.

Apply H6 with x3.

Apply andI with prim1 x3 (4ae4a.. (4ae4a.. 4a7ef..)), ∃ x6 . x2 = aae7a.. x3 x6 leaving 2 subgoals.

The subproof is completed by applying H2.

Let x6 of type ο be given.

Assume H7: ∀ x7 . x2 = aae7a.. x3 x7 ⟶ x6.

Apply H7 with x4.

The subproof is completed by applying H5.

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Proofgold Proof