simp only [evalPrimeFfToNot]
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
phi_ih :
  ↑phi.primeSet ⊆ Δ_U →
    IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
  ↑psi.primeSet ⊆ Δ_U →
    IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
phi_ih :
  ↑phi.primeSet ⊆ Δ_U →
    IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
  ↑psi.primeSet ⊆ Δ_U →
    IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
    (if evalPrime V (phi.imp_ psi) then phi.imp_ psi else (phi.imp_ psi).not_)