simp only [evalPrime]
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.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_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
    (if evalPrime V phi.not_ then phi.not_ else phi.not_.not_)
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.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_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
    (if ¬evalPrime V phi then phi.not_ else phi.not_.not_)