Theorem: FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn
Proof state before:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : admitsAux v u binders (pred_const_ a✝¹ a✝)
h2 : isFreeIn v (pred_const_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders

case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : admitsAux v u binders (pred_var_ a✝¹ a✝)
h2 : isFreeIn v (pred_var_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : admitsAux v u binders (eq_ a✝¹ a✝)
h2 : isFreeIn v (eq_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders

case true_
v u : VarName
binders : Finset VarName
h1 : admitsAux v u binders true_
h2 : isFreeIn v true_
h3 : v ∉ binders
⊢ u ∉ binders

case false_
v u : VarName
binders : Finset VarName
h1 : admitsAux v u binders false_
h2 : isFreeIn v false_
h3 : v ∉ binders
⊢ u ∉ binders

case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝.not_
h2 : isFreeIn v a✝.not_
h3 : v ∉ binders
⊢ u ∉ binders

case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (a✝¹.imp_ a✝)
h2 : isFreeIn v (a✝¹.imp_ a✝)
h3 : v ∉ binders
⊢ u ∉ binders

case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (a✝¹.and_ a✝)
h2 : isFreeIn v (a✝¹.and_ a✝)
h3 : v ∉ binders
⊢ u ∉ binders

case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (a✝¹.or_ a✝)
h2 : isFreeIn v (a✝¹.or_ a✝)
h3 : v ∉ binders
⊢ u ∉ binders

case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (a✝¹.iff_ a✝)
h2 : isFreeIn v (a✝¹.iff_ a✝)
h3 : v ∉ binders
⊢ u ∉ binders

case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (forall_ a✝¹ a✝)
h2 : isFreeIn v (forall_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders

case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (exists_ a✝¹ a✝)
h2 : isFreeIn v (exists_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders

case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : admitsAux v u binders (def_ a✝¹ a✝)
h2 : isFreeIn v (def_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
Tactic:
all_goals
  simp only [admitsAux] at h1
  simp only [isFreeIn] at h2
Proof state after:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
h2 : v ∈ a✝
h3 : v ∉ binders
⊢ u ∉ binders

case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
h2 : v ∈ a✝
h3 : v ∉ binders
⊢ u ∉ binders

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders
h2 : v = a✝¹ ∨ v = a✝
h3 : v ∉ binders
⊢ u ∉ binders

case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝
h2 : isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders

case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders

case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders

case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders

case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders

case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {a✝¹}) a✝
h2 : ¬v = a✝¹ ∧ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders

case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {a✝¹}) a✝
h2 : ¬v = a✝¹ ∧ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders

case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
h2 : v ∈ a✝
h3 : v ∉ binders
⊢ u ∉ binders