Theorem cbv1

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See cbv1v with disjoint variable conditions, not depending on ax-13 . (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.)

	Ref	Expression
Hypotheses	cbv1.1	$$⊢Ⅎx\phi$$
	cbv1.2	$$⊢Ⅎy\phi$$
	cbv1.3	$$⊢\phi \to Ⅎy\psi$$
	cbv1.4	$$⊢\phi \to Ⅎx\chi$$
	cbv1.5	$$⊢\phi \to \left(x=y\to \left(\psi \to \chi \right)\right)$$
Assertion	cbv1	$$⊢\phi \to \left(\forall x\psi \to \forall y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbv1.1	$$⊢Ⅎx\phi$$
2		cbv1.2	$$⊢Ⅎy\phi$$
3		cbv1.3	$$⊢\phi \to Ⅎy\psi$$
4		cbv1.4	$$⊢\phi \to Ⅎx\chi$$
5		cbv1.5	$$⊢\phi \to \left(x=y\to \left(\psi \to \chi \right)\right)$$
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	cbv3	$$⊢\forall x\left(\phi \to \psi \right)\to \forall y\left(\phi \to \chi \right)$$
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	$$⊢\phi \to \left(\forall x\psi \to \forall y\chi \right)$$