Theorem bj-nnf-cbvali

Description: Compared with bj-nnf-cbvaliv , replacing the DV condition on y , ps with the nonfreeness condition requires ax-11 . (Contributed by BJ, 4-Apr-2026)

	Ref	Expression
Hypotheses	bj-nnf-cbvali.nf0	$$⊢\phi \to \forall x\phi$$
	bj-nnf-cbvali.nf1	$$⊢\phi \to \forall y\phi$$
	bj-nnf-cbvali.ps	$$⊢\phi \to Ⅎ\text{'}y\psi$$
	bj-nnf-cbvali.ch	$$⊢\phi \to Ⅎ\text{'}x\chi$$
	bj-nnf-cbvali.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \to \chi \right)$$
Assertion	bj-nnf-cbvali	$$⊢\phi \to \left(\forall x\psi \to \forall y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-cbvali.nf0	$$⊢\phi \to \forall x\phi$$
2		bj-nnf-cbvali.nf1	$$⊢\phi \to \forall y\phi$$
3		bj-nnf-cbvali.ps	$$⊢\phi \to Ⅎ\text{'}y\psi$$
4		bj-nnf-cbvali.ch	$$⊢\phi \to Ⅎ\text{'}x\chi$$
5		bj-nnf-cbvali.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \to \chi \right)$$
6	3	bj-nnfad	$$⊢\phi \to \left(\psi \to \forall y\psi \right)$$
7	1 6	hbald	$$⊢\phi \to \left(\forall x\psi \to \forall y\forall x\psi \right)$$
8	1 4 5	bj-nnf-spim	$$⊢\phi \to \left(\forall x\psi \to \chi \right)$$
9	2 7 8	bj-alrimd	$$⊢\phi \to \left(\forall x\psi \to \forall y\chi \right)$$