Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Let x3 of type ι be given.

Let x4 of type ι be given.

Claim L1: ∃ x5 . and (prim1 x5 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (∃ x6 . and (prim1 x6 (If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) x1 (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) ... ...)))) ...)

...

Apply exandE_i with λ x5 . prim1 x5 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))), λ x5 . ∃ x6 . and (prim1 x6 (If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) x1 (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 x3)))) (x4 = aae7a.. x5 x6), ∃ x5 . and (prim1 x5 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (∃ x6 . x4 = aae7a.. x5 x6) leaving 2 subgoals.

The subproof is completed by applying L1.

Let x5 of type ι be given.

Assume H3: ∃ x6 . and (prim1 x6 (If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) x1 (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 x3)))) (x4 = aae7a.. x5 x6).

Apply exandE_i with λ x6 . prim1 x6 (If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) x1 (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 x3))), λ x6 . x4 = aae7a.. x5 x6, ∃ x6 . and (prim1 x6 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (∃ x7 . x4 = aae7a.. x6 x7) leaving 2 subgoals.

The subproof is completed by applying H3.

Let x6 of type ι be given.

Assume H4: prim1 x6 (If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) x1 (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 x3))).

Assume H5: x4 = aae7a.. x5 x6.

Let x7 of type ο be given.

Assume H6: ∀ x8 . and (prim1 x8 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (∃ x9 . x4 = aae7a.. x8 x9) ⟶ x7.

Apply H6 with x5.

Apply andI with prim1 x5 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))), ∃ x8 . x4 = aae7a.. x5 x8 leaving 2 subgoals.

The subproof is completed by applying H2.

Let x8 of type ο be given.

Assume H7: ∀ x9 . x4 = aae7a.. x5 x9 ⟶ x8.

Apply H7 with x6.

The subproof is completed by applying H5.

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Proofgold Proof