Theorem bj-nnf-cbval

Description: Compared with cbvalv1 , this saves ax-12 . (Contributed by BJ, 4-Apr-2026)

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

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-cbval.nf0	$$⊢\phi \to \forall x\phi$$
2		bj-nnf-cbval.nf1	$$⊢\phi \to \forall y\phi$$
3		bj-nnf-cbval.ps	$$⊢\phi \to Ⅎ\text{'}y\psi$$
4		bj-nnf-cbval.ch	$$⊢\phi \to Ⅎ\text{'}x\chi$$
5		bj-nnf-cbval.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
6	5	biimpd	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \to \chi \right)$$
7	1 2 3 4 6	bj-nnf-cbvali	$$⊢\phi \to \left(\forall x\psi \to \forall y\chi \right)$$
8		equcomi	$$⊢y=x\to x=y$$
9	8 5	sylan2	$$⊢\left(\phi \wedge y=x\right)\to \left(\psi \leftrightarrow \chi \right)$$
10	9	biimprd	$$⊢\left(\phi \wedge y=x\right)\to \left(\chi \to \psi \right)$$
11	2 1 4 3 10	bj-nnf-cbvali	$$⊢\phi \to \left(\forall y\chi \to \forall x\psi \right)$$
12	7 11	impbid	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$