have ℓ_in_ψ_in_Rcond : ∃ψ ∈ Rcond, ℓ ∈ voc ψ := by
  let voc_res_ss_Rcond := localRule_does_not_increase_vocab_R rule res res_in_ress
  simp at voc_res_ss_Rcond
  let ℓ_in_voc_Rcond := voc_res_ss_Rcond Rφ Rφ_in_res_R ℓ_in_Rφ
  simp at ℓ_in_voc_Rcond
  exact ℓ_in_voc_Rcond
case right.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
Rφ : Formula
Rφ_in_cR : Rφ ∈ c.2
ℓ_in_Rφ : ℓ ∈ vocabOfFormula Rφ
not_Rφ_in_R : Rφ ∉ R
Rφ_in_res : ∃ res ∈ ress, Rφ ∈ res.2
res : SubPair
res_in_ress : res ∈ ress
Rφ_in_res_R : Rφ ∈ res.2
⊢ ∃ a ∈ R, ℓ ∈ vocabOfFormula a
case right.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
Rφ : Formula
Rφ_in_cR : Rφ ∈ c.2
ℓ_in_Rφ : ℓ ∈ vocabOfFormula Rφ
not_Rφ_in_R : Rφ ∉ R
Rφ_in_res : ∃ res ∈ ress, Rφ ∈ res.2
res : SubPair
res_in_ress : res ∈ ress
Rφ_in_res_R : Rφ ∈ res.2
ℓ_in_ψ_in_Rcond : ∃ ψ ∈ Rcond, ℓ ∈ voc ψ
⊢ ∃ a ∈ R, ℓ ∈ vocabOfFormula a