Let p, q and α be given.

Assume Ha.

Let β be given.

Assume Hb H1.

Apply H1 to the current goal.

Let γ be given.

Assume H2.

Apply H2 to the current goal.

Assume H2: γ ∈ β.

Assume H3.

We will prove ∃β ∈ α, PNoEq_ β p q ∧ ¬ p β ∧ q β.

We use γ to witness the existential quantifier.

Apply andI to the current goal.

We will prove γ ∈ α.

Apply Ha to the current goal.

Assume Ha1 _.

An exact proof term for the current goal is Ha1 β Hb γ H2.

An exact proof term for the current goal is H3.

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