simp only [Holds]
case exists_
D : Type
I : Interpretation D
E : Env
σ : VarName → VarName
h1 : Function.Injective σ
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (V ∘ σ) E a✝ ↔ Holds D I V E (replaceAll σ a✝)
V : VarAssignment D
⊢ Holds D I (V ∘ σ) E (exists_ a✝¹ a✝) ↔ Holds D I V E (exists_ (σ a✝¹) (replaceAll σ a✝))
case exists_
D : Type
I : Interpretation D
E : Env
σ : VarName → VarName
h1 : Function.Injective σ
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (V ∘ σ) E a✝ ↔ Holds D I V E (replaceAll σ a✝)
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE (V ∘ σ) a✝¹ d) E a✝) ↔
    ∃ d, Holds D I (Function.updateITE V (σ a✝¹) d) E (replaceAll σ a✝)