obtain s1 := substitution_theorem D I V E τ F h1
F : Formula
τ : PredName → ℕ → List VarName × Formula
h1 : admits τ F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E (replace τ F)
F : Formula
τ : PredName → ℕ → List VarName × Formula
h1 : admits τ F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
s1 :
  Holds D
      { nonempty := ⋯, pred_const_ := I.pred_const_,
        pred_var_ := fun X ds =>
          let zs := (τ X ds.length).1;
          let H := (τ X ds.length).2;
          if ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ X ds }
      V E F ↔
    Holds D I V E (replace τ F)
⊢ Holds D I V E (replace τ F)