Theorem cbvralsv 2589

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 1462	. . 3 |- F/ z ph
2		nfs1v 1857	. . 3 |- F/ x [ z / x ] ph
3		sbequ12 1695	. . 3 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
4	1, 2, 3	cbvral 2574	. 2 |- ( A. x e. A ph <-> A. z e. A [ z / x ] ph )
5		nfv 1462	. . . 4 |- F/ y ph
6	5	nfsb 1864	. . 3 |- F/ y [ z / x ] ph
7		nfv 1462	. . 3 |- F/ z [ y / x ] ph
8		sbequ 1762	. . 3 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
9	6, 7, 8	cbvral 2574	. 2 |- ( A. z e. A [ z / x ] ph <-> A. y e. A [ y / x ] ph )
10	4, 9	bitri 182	1 |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )

Syntax hints:   <-> wb 103   [wsb 1686   A.wral 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354

This theorem is referenced by:  sbralie  2591  rspsbc  2897  ralxpf  4504