Theorem cbvralsv 2708

Description: Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)

Assertion

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

Distinct variable groups:   x, A   y, A   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem cbvralsv

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1516	. . 3 |- F/ z ph
2		nfs1v 1927	. . 3 |- F/ x [ z / x ] ph
3		sbequ12 1759	. . 3 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
4	1, 2, 3	cbvral 2688	. 2 |- ( A. x e. A ph <-> A. z e. A [ z / x ] ph )
5		nfv 1516	. . . 4 |- F/ y ph
6	5	nfsb 1934	. . 3 |- F/ y [ z / x ] ph
7		nfv 1516	. . 3 |- F/ z [ y / x ] ph
8		sbequ 1828	. . 3 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
9	6, 7, 8	cbvral 2688	. 2 |- ( A. z e. A [ z / x ] ph <-> A. y e. A [ y / x ] ph )
10	4, 9	bitri 183	1 |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )

Syntax hints:   <-> wb 104   [wsb 1750   A.wral 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449

This theorem is referenced by:  sbralie  2710  rspsbc  3033  ralxpf  4750