Theorem cbvralsv

Description: Change bound variable by using a substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvralsvw when possible. (Contributed by NM, 20-Nov-2005) (Revised by Andrew Salmon, 11-Jul-2011) (New usage is discouraged.)

	Ref	Expression
Assertion	cbvralsv	$$⊢\forall x\in A\phi \leftrightarrow \forall y\in A\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		nfv	$$⊢Ⅎz\phi$$
2		nfs1v	$$⊢Ⅎx\left[z/x\right]\phi$$
3		sbequ12	$$⊢x=z\to \left(\phi \leftrightarrow \left[z/x\right]\phi \right)$$
4	1 2 3	cbvral	$$⊢\forall x\in A\phi \leftrightarrow \forall z\in A\left[z/x\right]\phi$$
5		nfv	$$⊢Ⅎy\phi$$
6	5	nfsb	$$⊢Ⅎy\left[z/x\right]\phi$$
7		nfv	$$⊢Ⅎz\left[y/x\right]\phi$$
8		sbequ	$$⊢z=y\to \left(\left[z/x\right]\phi \leftrightarrow \left[y/x\right]\phi \right)$$
9	6 7 8	cbvral	$$⊢\forall z\in A\left[z/x\right]\phi \leftrightarrow \forall y\in A\left[y/x\right]\phi$$
10	4 9	bitri	$$⊢\forall x\in A\phi \leftrightarrow \forall y\in A\left[y/x\right]\phi$$