Theorem cbvald

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim . Usage of this theorem is discouraged because it depends on ax-13 . See cbvaldw for a version with x , y disjoint, not depending on ax-13 . (Contributed by NM, 2-Jan-2002) (Revised by Mario Carneiro, 6-Oct-2016) (Revised by Wolf Lammen, 13-May-2018) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvald.1	$$⊢Ⅎy\phi$$
2		cbvald.2	$$⊢\phi \to Ⅎy\psi$$
3		cbvald.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
4		nfv	$$⊢Ⅎx\phi$$
5		nfvd	$$⊢\phi \to Ⅎx\chi$$
6	4 1 2 5 3	cbv2	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$