Theorem cbv1v

Description: Rule used to change bound variables, using implicit substitution. Version of cbv1 with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 16-Jun-2019)

	Ref	Expression
Hypotheses	cbv1v.1	$$⊢Ⅎx\phi$$
	cbv1v.2	$$⊢Ⅎy\phi$$
	cbv1v.3	$$⊢\phi \to Ⅎy\psi$$
	cbv1v.4	$$⊢\phi \to Ⅎx\chi$$
	cbv1v.5	$$⊢\phi \to \left(x=y\to \left(\psi \to \chi \right)\right)$$
Assertion	cbv1v	$$⊢\phi \to \left(\forall x\psi \to \forall y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbv1v.1	$$⊢Ⅎx\phi$$
2		cbv1v.2	$$⊢Ⅎy\phi$$
3		cbv1v.3	$$⊢\phi \to Ⅎy\psi$$
4		cbv1v.4	$$⊢\phi \to Ⅎx\chi$$
5		cbv1v.5	$$⊢\phi \to \left(x=y\to \left(\psi \to \chi \right)\right)$$
6	2 3	nfim1	$$⊢Ⅎy\left(\phi \to \psi \right)$$
7	1 4	nfim1	$$⊢Ⅎx\left(\phi \to \chi \right)$$
8	5	com12	$$⊢x=y\to \left(\phi \to \left(\psi \to \chi \right)\right)$$
9	8	a2d	$$⊢x=y\to \left(\left(\phi \to \psi \right)\to \left(\phi \to \chi \right)\right)$$
10	6 7 9	cbv3v	$$⊢\forall x\left(\phi \to \psi \right)\to \forall y\left(\phi \to \chi \right)$$
11	1	19.21	$$⊢\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)$$
12	2	19.21	$$⊢\forall y\left(\phi \to \chi \right)\leftrightarrow \left(\phi \to \forall y\chi \right)$$
13	10 11 12	3imtr3i	$$⊢\left(\phi \to \forall x\psi \right)\to \left(\phi \to \forall y\chi \right)$$
14	13	pm2.86i	$$⊢\phi \to \left(\forall x\psi \to \forall y\chi \right)$$