Let L and R be given.
Assume HLR.
Let α be given.
Assume Ha HaL HaR.
Let β be given.
Assume H1.
Apply H1 to the current goal.
Assume H1.
Apply H1 to the current goal.
Let p be given.
We use β to witness the existential quantifier.
Apply andI to the current goal.
We use p to witness the existential quantifier.
An exact proof term for the current goal is Hp.
Apply L1 to the current goal.
Let β be given.
Assume H1.
Apply H1 to the current goal.
Assume H1.
Apply H1 to the current goal.
Apply H2 to the current goal.
Let p be given.
We use β to witness the existential quantifier.
Apply andI to the current goal.
We use
(λx ⇒ x ∈ β ∧ p x) to
witness the existential quantifier.
Apply andI to the current goal.
Apply and3I to the current goal.
An exact proof term for the current goal is H1.
We will
prove PNoEq_ β p (λx ⇒ x ∈ β ∧ p x).
Let x be given.
We will
prove p x ↔ x ∈ β ∧ p x.
Apply iffI to the current goal.
Assume H4.
Apply andI to the current goal.
An exact proof term for the current goal is Hx.
An exact proof term for the current goal is H4.
Assume H4.
Apply H4 to the current goal.
Assume _ H5.
An exact proof term for the current goal is H5.
An exact proof term for the current goal is Hp.
Let γ be given.
Let q be given.
Apply H3 γ Hc to the current goal.
We use q to witness the existential quantifier.
An exact proof term for the current goal is H4.
We will
prove ∀x, x ∉ β → ¬ (x ∈ β ∧ p x).
Let x be given.
Assume Hx.
Assume H4.
Apply H4 to the current goal.
Assume H5 _.
An exact proof term for the current goal is Hx H5.
Let q and r be given.
Apply Hq to the current goal.
Assume Hq1 Hq2.
Apply Hr to the current goal.
Assume Hr1 Hr2.
Let x be given.
Apply xm (x ∈ β) to the current goal.
Apply Hr1 to the current goal.
Assume Hr1a.
Apply Hr1a to the current goal.
Assume _.
Assume _.
Apply iffI to the current goal.
Assume H5: q x.
Apply Hq2 x H4 to the current goal.
An exact proof term for the current goal is H5.
Assume H5: r x.
Apply Hr2 x H4 to the current goal.
An exact proof term for the current goal is H5.
∎