Theorem cbv2OLD

Description: Obsolete version of cbv2 as of 10-Sep-2023. (Contributed by NM, 5-Aug-1993) (Revised by Mario Carneiro, 3-Oct-2016) Format hypotheses to common style. (Revised by Wolf Lammen, 13-May-2018) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	cbv2OLD.1	$$⊢Ⅎx\phi$$
	cbv2OLD.2	$$⊢Ⅎy\phi$$
	cbv2OLD.3	$$⊢\phi \to Ⅎy\psi$$
	cbv2OLD.4	$$⊢\phi \to Ⅎx\chi$$
	cbv2OLD.5	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
Assertion	cbv2OLD	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbv2OLD.1	$$⊢Ⅎx\phi$$
2		cbv2OLD.2	$$⊢Ⅎy\phi$$
3		cbv2OLD.3	$$⊢\phi \to Ⅎy\psi$$
4		cbv2OLD.4	$$⊢\phi \to Ⅎx\chi$$
5		cbv2OLD.5	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
6	2	nf5ri	$$⊢\phi \to \forall y\phi$$
7	1 6	alrimi	$$⊢\phi \to \forall x\forall y\phi$$
8	3	nf5rd	$$⊢\phi \to \left(\psi \to \forall y\psi \right)$$
9	4	nf5rd	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
10	8 9 5	cbv2h	$$⊢\forall x\forall y\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$
11	7 10	syl	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$