apply T_13_6
case a.h1
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
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_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsProof (phi.not_.imp_ (phi.imp_ psi))

case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
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_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
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_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_