Proposition from_eq_in_equalizer_partial_setoid_ex {Γ : ty H} {Δ : form Γ} (t₁ t₂ : tm Γ equalizer_partial_setoid) (p : Δ ⊢ t₁ ~ t₂) : Δ ⊢ ∃h (φ [ ⟨ t₁ [ π₁ (tm_var _) ]tm , π₂ (tm_var _) ⟩ ] ∧ ψ [ ⟨ t₁ [ π₁ (tm_var _) ]tm , π₂ (tm_var _) ⟩ ]). Proof. refine (hyperdoctrine_cut p _). unfold partial_setoid_formula ; cbn. unfold equalizer_per_form. rewrite conj_subst. use weaken_right. simplify. apply hyperdoctrine_hyp. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Hyperdoctrines.Hyperdoctrine. Require Import UniMath.CategoryTheory.Hyperdoctrines.FirstOrderHyperdoctrine. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.PERs. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.PERMorphisms. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.PERCategory.