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Proofgold Proof

pf
Let x0 of type ((((ιο) → ο) → ο) → ο) → ο be given.
Assume H0: ∃ x1 : (((ι → ο) → ο) → ο) → ο . x0 x1.
Assume H1: ∀ x1 x2 : (((ι → ο) → ο) → ο) → ο . x0 x1x0 x2x1 = x2.
Apply H0 with x0 (Descr_Vo4 x0).
Let x1 of type (((ιο) → ο) → ο) → ο be given.
Assume H2: x0 x1.
Claim L3: x1 = Descr_Vo4 x0
Apply functional extensionality with x1, Descr_Vo4 x0.
Let x2 of type ((ιο) → ο) → ο be given.
Apply prop_ext_2 with x1 x2, Descr_Vo4 x0 x2 leaving 2 subgoals.
Assume H3: x1 x2.
Let x3 of type (((ιο) → ο) → ο) → ο be given.
Assume H4: x0 x3.
Apply H1 with x1, x3, λ x4 x5 : (((ι → ο) → ο) → ο) → ο . x4 x2 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
The subproof is completed by applying H3.
Assume H3: Descr_Vo4 x0 x2.
Apply H3 with x1.
The subproof is completed by applying H2.
Apply L3 with λ x2 x3 : (((ι → ο) → ο) → ο) → ο . x0 x2.
The subproof is completed by applying H2.