Theorem cbv2w

Description: Rule used to change bound variables, using implicit substitution. Version of cbv2 with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbv2w.1	$$⊢Ⅎx\phi$$
	cbv2w.2	$$⊢Ⅎy\phi$$
	cbv2w.3	$$⊢\phi \to Ⅎy\psi$$
	cbv2w.4	$$⊢\phi \to Ⅎx\chi$$
	cbv2w.5	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
Assertion	cbv2w	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbv2w.1	$$⊢Ⅎx\phi$$
2		cbv2w.2	$$⊢Ⅎy\phi$$
3		cbv2w.3	$$⊢\phi \to Ⅎy\psi$$
4		cbv2w.4	$$⊢\phi \to Ⅎx\chi$$
5		cbv2w.5	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
6		biimp	$$⊢\left(\psi \leftrightarrow \chi \right)\to \left(\psi \to \chi \right)$$
7	5 6	syl6	$$⊢\phi \to \left(x=y\to \left(\psi \to \chi \right)\right)$$
8	1 2 3 4 7	cbv1v	$$⊢\phi \to \left(\forall x\psi \to \forall y\chi \right)$$
9		equcomi	$$⊢y=x\to x=y$$
10		biimpr	$$⊢\left(\psi \leftrightarrow \chi \right)\to \left(\chi \to \psi \right)$$
11	9 5 10	syl56	$$⊢\phi \to \left(y=x\to \left(\chi \to \psi \right)\right)$$
12	2 1 4 3 11	cbv1v	$$⊢\phi \to \left(\forall y\chi \to \forall x\psi \right)$$
13	8 12	impbid	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$