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	$⊢ ((φ \leftrightarrow ψ) \land \forall x (φ \leftrightarrow ψ)) \to (Ⅎ' x φ \leftrightarrow Ⅎ' x ψ)$

Proof

Step	Hyp	Ref	Expression
1		bj-hbyfrbi	$⊢ ((φ \leftrightarrow ψ) \land \forall x (φ \leftrightarrow ψ)) \to ((\exists x φ \to φ) \leftrightarrow (\exists x ψ \to ψ))$
2		bj-hbxfrbi	$⊢ ((φ \leftrightarrow ψ) \land \forall x (φ \leftrightarrow ψ)) \to ((φ \to \forall x φ) \leftrightarrow (ψ \to \forall x ψ))$
3	1 2	anbi12d	$⊢ ((φ \leftrightarrow ψ) \land \forall x (φ \leftrightarrow ψ)) \to (((\exists x φ \to φ) \land (φ \to \forall x φ)) \leftrightarrow ((\exists x ψ \to ψ) \land (ψ \to \forall x ψ)))$
4		df-bj-nnf	$⊢ Ⅎ' x φ \leftrightarrow ((\exists x φ \to φ) \land (φ \to \forall x φ))$
5		df-bj-nnf	$⊢ Ⅎ' x ψ \leftrightarrow ((\exists x ψ \to ψ) \land (ψ \to \forall x ψ))$
6	3 4 5	3bitr4g	$⊢ ((φ \leftrightarrow ψ) \land \forall x (φ \leftrightarrow ψ)) \to (Ⅎ' x φ \leftrightarrow Ⅎ' x ψ)$

Metamath Proof Explorer

Theorem bj-nnfbi

Proof