apply Holds_coincide_Var
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
    fastAdmitsAux v t binders F →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
⊢ Holds D I
      (Function.updateListITE (fun c => if c = v then V' t else V c) hd.args
        (List.map (fun c => if c = v then V' t else V c) xs))
      tl hd.q ↔
    Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if v = x then t else x) xs))) tl hd.q
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
    fastAdmitsAux v t binders F →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
⊢ ∀ (v_1 : VarName),
    isFreeIn v_1 hd.q →
      Function.updateListITE (fun c => if c = v then V' t else V c) hd.args
          (List.map (fun c => if c = v then V' t else V c) xs) v_1 =
        Function.updateListITE V hd.args (List.map V (List.map (fun x => if v = x then t else x) xs)) v_1