Theorem cbval2v

Description: Rule used to change bound variables, using implicit substitution. Version of cbval2 with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 22-Dec-2003) (Revised by BJ, 16-Jun-2019) (Proof shortened by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbval2v.1	$$⊢Ⅎz\phi$$
	cbval2v.2	$$⊢Ⅎw\phi$$
	cbval2v.3	$$⊢Ⅎx\psi$$
	cbval2v.4	$$⊢Ⅎy\psi$$
	cbval2v.5	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbval2v	$$⊢\forall x\forall y\phi \leftrightarrow \forall z\forall w\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbval2v.1	$$⊢Ⅎz\phi$$
2		cbval2v.2	$$⊢Ⅎw\phi$$
3		cbval2v.3	$$⊢Ⅎx\psi$$
4		cbval2v.4	$$⊢Ⅎy\psi$$
5		cbval2v.5	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
6	1	nfal	$$⊢Ⅎz\forall y\phi$$
7	3	nfal	$$⊢Ⅎx\forall w\psi$$
8		nfv	$$⊢Ⅎyx=z$$
9		nfv	$$⊢Ⅎwx=z$$
10	2	a1i	$$⊢x=z\to Ⅎw\phi$$
11	4	a1i	$$⊢x=z\to Ⅎy\psi$$
12	5	ex	$$⊢x=z\to \left(y=w\to \left(\phi \leftrightarrow \psi \right)\right)$$
13	8 9 10 11 12	cbv2w	$$⊢x=z\to \left(\forall y\phi \leftrightarrow \forall w\psi \right)$$
14	6 7 13	cbvalv1	$$⊢\forall x\forall y\phi \leftrightarrow \forall z\forall w\psi$$