case rename_forall_ h1_phi h1_x h1_y h1_1 h1_2 | rename_exists_ h1_phi h1_x h1_y h1_1 h1_2 =>
  simp only [Holds]
  first | apply forall_congr' | apply exists_congr
  intro d
  exact replace_empty_Holds D I V E h1_x h1_y h1_phi d h1_1 h1_2
D : Type
I : Interpretation D
E : Env
F F' h1_phi : Formula
h1_x h1_y : VarName
h1_1 : ¬isFreeIn h1_y h1_phi
h1_2 : ¬isBoundIn h1_y h1_phi
V : VarAssignment D
⊢ Holds D I V E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_y (Sub.Var.One.Rec.fastReplaceFree h1_x h1_y h1_phi))
no goals