simp
case a
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ (if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_) ∈
    (fun a => if evalPrime V a then a else a.not_) '' Δ_U
case a
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ ∃ x_1 ∈ Δ_U,
    (if evalPrime V x_1 then x_1 else x_1.not_) =
      if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_