have ℓ_in_ψ_in_Lcond : ∃ψ ∈ Lcond, ℓ ∈ voc ψ := by
  let voc_res_ss_Lcond := localRule_does_not_increase_vocab_L rule res res_in_ress
  simp at voc_res_ss_Lcond
  let ℓ_in_voc_Lcond := voc_res_ss_Lcond Lφ Lφ_in_res_L ℓ_in_Lφ
  simp at ℓ_in_voc_Lcond
  exact ℓ_in_voc_Lcond
case left.right
C : List TNode
L R : Finset Formula
ruleA : LocalRuleApp (L, R) C
ress : List SubPair
Lcond Rcond : Finset Formula
rule : LocalRule (Lcond, Rcond) ress
hC : C = applyLocalRule rule (L, R)
preproof : Lcond ⊆ L ∧ Rcond ⊆ R
c : TNode
cinC : c ∈ C
ℓ : Char
ℓ_in_c : (∃ a ∈ c.1, ℓ ∈ vocabOfFormula a) ∧ ∃ a ∈ c.2, ℓ ∈ vocabOfFormula a
Lφ : Formula
Lφ_in_cL : Lφ ∈ c.1
ℓ_in_Lφ : ℓ ∈ vocabOfFormula Lφ
not_Lφ_in_L : Lφ ∉ L
Lφ_in_res : ∃ res ∈ ress, Lφ ∈ res.1
res : SubPair
res_in_ress : res ∈ ress
Lφ_in_res_L : Lφ ∈ res.1
⊢ ∃ a ∈ L, ℓ ∈ vocabOfFormula a
case left.right
C : List TNode
L R : Finset Formula
ruleA : LocalRuleApp (L, R) C
ress : List SubPair
Lcond Rcond : Finset Formula
rule : LocalRule (Lcond, Rcond) ress
hC : C = applyLocalRule rule (L, R)
preproof : Lcond ⊆ L ∧ Rcond ⊆ R
c : TNode
cinC : c ∈ C
ℓ : Char
ℓ_in_c : (∃ a ∈ c.1, ℓ ∈ vocabOfFormula a) ∧ ∃ a ∈ c.2, ℓ ∈ vocabOfFormula a
Lφ : Formula
Lφ_in_cL : Lφ ∈ c.1
ℓ_in_Lφ : ℓ ∈ vocabOfFormula Lφ
not_Lφ_in_L : Lφ ∉ L
Lφ_in_res : ∃ res ∈ ress, Lφ ∈ res.1
res : SubPair
res_in_ress : res ∈ ress
Lφ_in_res_L : Lφ ∈ res.1
ℓ_in_ψ_in_Lcond : ∃ ψ ∈ Lcond, ℓ ∈ voc ψ
⊢ ∃ a ∈ L, ℓ ∈ vocabOfFormula a