congr! 1
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ exists_ x (replaceFreeAux σ (binders ∪ {x}) phi) = exists_ x (replaceFreeAux σ' (binders ∪ {x}) phi)
case h.e'_2
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ (binders ∪ {x}) phi = replaceFreeAux σ' (binders ∪ {x}) phi