simp only [Holds] at ih
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
ih :
  ∀ (V : VarAssignment D) (F : Formula),
    Holds D I V tl (sub τ F) ↔
      Holds D
        { nonempty := ⋯, pred_const_ := I.pred_const_,
          pred_var_ := fun P ds => if ds = [] then Holds D I V tl (pred_var_ (τ P) []) else I.pred_var_ P ds }
        V tl F
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
    Holds D
      { nonempty := ⋯, pred_const_ := I.pred_const_,
        pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
      V (hd :: tl) (def_ X xs)
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
  ∀ (V : VarAssignment D) (F : Formula),
    Holds D I V tl (sub τ F) ↔
      Holds D
        { nonempty := ⋯, pred_const_ := I.pred_const_,
          pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
        V tl F
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
    Holds D
      { nonempty := ⋯, pred_const_ := I.pred_const_,
        pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
      V (hd :: tl) (def_ X xs)