Theorem cbval2

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbval2v if possible. (Contributed by NM, 22-Dec-2003) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 11-Sep-2023) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbval2.1	$$⊢Ⅎz\phi$$
	cbval2.2	$$⊢Ⅎw\phi$$
	cbval2.3	$$⊢Ⅎx\psi$$
	cbval2.4	$$⊢Ⅎy\psi$$
	cbval2.5	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbval2	$$⊢\forall x\forall y\phi \leftrightarrow \forall z\forall w\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbval2.1	$$⊢Ⅎz\phi$$
2		cbval2.2	$$⊢Ⅎw\phi$$
3		cbval2.3	$$⊢Ⅎx\psi$$
4		cbval2.4	$$⊢Ⅎy\psi$$
5		cbval2.5	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
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	cbv2	$$⊢x=z\to \left(\forall y\phi \leftrightarrow \forall w\psi \right)$$
14	6 7 13	cbval	$$⊢\forall x\forall y\phi \leftrightarrow \forall z\forall w\psi$$