intro d
case h
D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ ∀ (a : D),
    Holds D I (Function.updateITE V (free_ z) a ∘ Var.openList 0 [free_ z]) F ↔
      Holds D I (Function.updateITE V (free_ z) a) (Formula.openList 0 [free_ z] F)
case h
D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
⊢ Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F ↔
    Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F)