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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.
StepHypRef 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 ) )
41, 2, 3cbvral 2574 . 2  |-  ( A. x  e.  A  ph  <->  A. z  e.  A  [ z  /  x ] ph )
5 nfv 1462 . . . 4  |-  F/ y
ph
65nfsb 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 ) )
96, 7, 8cbvral 2574 . 2  |-  ( A. z  e.  A  [
z  /  x ] ph 
<-> 
A. y  e.  A  [ y  /  x ] ph )
104, 9bitri 182 1  |-  ( A. x  e.  A  ph  <->  A. y  e.  A  [ y  /  x ] ph )
Colors of variables: wff set class
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
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