Theorem bj-nnf-cbvaliv

Description: The only DV conditions are those saying that y is a fresh variable used to construct ch . (Contributed by BJ, 4-Apr-2026)

	Ref	Expression
Hypotheses	bj-nnf-cbvaliv.nf0	$$⊢\phi \to \forall x\phi$$
	bj-nnf-cbvaliv.nf	$$⊢\phi \to Ⅎ\text{'}x\chi$$
	bj-nnf-cbvaliv.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \to \chi \right)$$
Assertion	bj-nnf-cbvaliv	$$⊢\phi \to \left(\forall x\psi \to \forall y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-cbvaliv.nf0	$$⊢\phi \to \forall x\phi$$
2		bj-nnf-cbvaliv.nf	$$⊢\phi \to Ⅎ\text{'}x\chi$$
3		bj-nnf-cbvaliv.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \to \chi \right)$$
4		ax-5	$$⊢\phi \to \forall y\phi$$
5		ax-5	$$⊢\forall x\psi \to \forall y\forall x\psi$$
6	1 2 3	bj-nnf-spim	$$⊢\phi \to \left(\forall x\psi \to \chi \right)$$
7	4 5 6	alrimdh	$$⊢\phi \to \left(\forall x\psi \to \forall y\chi \right)$$