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	|- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )

Proof

Step	Hyp	Ref	Expression
1		nfv	|- F/ z ph
2		nfs1v	|- F/ x [ z / x ] ph
3		sbequ12	|- ( x = z -> ( ph <-> [ z / x ] ph ) )
4	1 2 3	cbvral	|- ( A. x e. A ph <-> A. z e. A [ z / x ] ph )
5		nfv	|- F/ y ph
6	5	nfsb	|- F/ y [ z / x ] ph
7		nfv	|- F/ z [ y / x ] ph
8		sbequ	|- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
9	6 7 8	cbvral	|- ( A. z e. A [ z / x ] ph <-> A. y e. A [ y / x ] ph )
10	4 9	bitri	|- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )