congr! 1
phi psi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ (fastReplaceFree σ phi).iff_ (fastReplaceFree σ psi) = (fastReplaceFree σ' phi).iff_ (fastReplaceFree σ' psi)
case h.e'_1
phi psi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ fastReplaceFree σ phi = fastReplaceFree σ' phi

case h.e'_2
phi psi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ fastReplaceFree σ psi = fastReplaceFree σ' psi