Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι be given.
Apply iffI with
prim1 x2 (0fc90.. x0 (λ x3 . x1 x3)),
∃ x3 . and (prim1 x3 x0) (∃ x4 . and (prim1 x4 (x1 x3)) (x2 = aae7a.. x3 x4)) leaving 2 subgoals.
The subproof is completed by applying unknownprop_d4dc73f3cbfe4c22363272ac418d035b8e77c49433b0870b96bee8fa9c46bfcf with x0, λ x3 . x1 x3, x2.
Apply exandE_i with
λ x3 . prim1 x3 x0,
λ x3 . ∃ x4 . and (prim1 x4 (x1 x3)) (x2 = aae7a.. x3 x4),
prim1 x2 (0fc90.. x0 (λ x3 . x1 x3)) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ι be given.
Apply exandE_i with
λ x4 . prim1 x4 (x1 x3),
λ x4 . x2 = aae7a.. x3 x4,
prim1 x2 (0fc90.. x0 (λ x4 . x1 x4)) leaving 2 subgoals.
The subproof is completed by applying H2.
Let x4 of type ι be given.
Assume H3:
prim1 x4 (x1 x3).
Apply H4 with
λ x5 x6 . prim1 x6 (0fc90.. x0 (λ x7 . x1 x7)).
Apply unknownprop_1f27075d0cd8d16888a609d68ca6246fb2307839dccadd646f85ab18bdcaae8e with
x0,
λ x5 . x1 x5,
x3,
x4 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.