induction vs using Finset.induction generalizing τ
ν : Type u_1
inst✝¹ inst✝ : DecidableEq ν
τ : PropAssignment ν
vs : Finset ν
τ' : PropAssignment ν
φ : PropFun ν
⊢ PropAssignment.setMany τ vs τ' ⊨ φ ↔ PropAssignment.setMany τ (vs ∩ semVars φ) τ' ⊨ φ
case empty
ν : Type u_1
inst✝¹ inst✝ : DecidableEq ν
τ' : PropAssignment ν
φ : PropFun ν
τ : PropAssignment ν
⊢ PropAssignment.setMany τ ∅ τ' ⊨ φ ↔ PropAssignment.setMany τ (∅ ∩ semVars φ) τ' ⊨ φ

case insert
ν : Type u_1
inst✝¹ inst✝ : DecidableEq ν
τ' : PropAssignment ν
φ : PropFun ν
a✝² : ν
s✝ : Finset ν
a✝¹ : a✝² ∉ s✝
a✝ : ∀ (τ : PropAssignment ν), PropAssignment.setMany τ s✝ τ' ⊨ φ ↔ PropAssignment.setMany τ (s✝ ∩ semVars φ) τ' ⊨ φ
τ : PropAssignment ν
⊢ PropAssignment.setMany τ (insert a✝² s✝) τ' ⊨ φ ↔ PropAssignment.setMany τ (insert a✝² s✝ ∩ semVars φ) τ' ⊨ φ