congr! 1
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ exists_ x (fastReplaceFree (Function.updateITE σ x x) phi) =
    exists_ x (fastReplaceFree (Function.updateITE σ' x x) phi)
case h.e'_2
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ fastReplaceFree (Function.updateITE σ x x) phi = fastReplaceFree (Function.updateITE σ' x x) phi