Metamath Proof Explorer

Theorem bj-nnfbi

Description: If two formulas are equivalent for all x , then nonfreeness of x in one of them is equivalent to nonfreeness in the other. Compare nfbiit . From this and bj-nnfim and bj-nnfnt , one can prove analogous nonfreeness conservation results for other propositional operators. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht ) in order not to require sp (modal T). (Contributed by BJ, 27-Aug-2023)

Ref Expression
Assertion bj-nnfbi ⊢ φ ↔ ψ ∧ ∀ x φ ↔ ψ → Ⅎ' x φ ↔ Ⅎ' x ψ

Proof

Step Hyp Ref Expression
1 bj-hbyfrbi ⊢ φ ↔ ψ ∧ ∀ x φ ↔ ψ → ∃ x φ → φ ↔ ∃ x ψ → ψ
2 bj-hbxfrbi ⊢ φ ↔ ψ ∧ ∀ x φ ↔ ψ → φ → ∀ x φ ↔ ψ → ∀ x ψ
3 1 2 anbi12d ⊢ φ ↔ ψ ∧ ∀ x φ ↔ ψ → ∃ x φ → φ ∧ φ → ∀ x φ ↔ ∃ x ψ → ψ ∧ ψ → ∀ x ψ
4 df-bj-nnf ⊢ Ⅎ' x φ ↔ ∃ x φ → φ ∧ φ → ∀ x φ
5 df-bj-nnf ⊢ Ⅎ' x ψ ↔ ∃ x ψ → ψ ∧ ψ → ∀ x ψ
6 3 4 5 3bitr4g ⊢ φ ↔ ψ ∧ ∀ x φ ↔ ψ → Ⅎ' x φ ↔ Ⅎ' x ψ