Theorem cbvralsv 2602

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 1467	. . 3 |- F/ z ph
2		nfs1v 1864	. . 3 |- F/ x [ z / x ] ph
3		sbequ12 1702	. . 3 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
4	1, 2, 3	cbvral 2587	. 2 |- ( A. x e. A ph <-> A. z e. A [ z / x ] ph )
5		nfv 1467	. . . 4 |- F/ y ph
6	5	nfsb 1871	. . 3 |- F/ y [ z / x ] ph
7		nfv 1467	. . 3 |- F/ z [ y / x ] ph
8		sbequ 1769	. . 3 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
9	6, 7, 8	cbvral 2587	. 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 1693   A.wral 2360

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365

This theorem is referenced by:  sbralie  2604  rspsbc  2922  ralxpf  4595