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Theorem cbvralsv 2663
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 1508 . . 3  |-  F/ z
ph
2 nfs1v 1910 . . 3  |-  F/ x [ z  /  x ] ph
3 sbequ12 1744 . . 3  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
41, 2, 3cbvral 2648 . 2  |-  ( A. x  e.  A  ph  <->  A. z  e.  A  [ z  /  x ] ph )
5 nfv 1508 . . . 4  |-  F/ y
ph
65nfsb 1917 . . 3  |-  F/ y [ z  /  x ] ph
7 nfv 1508 . . 3  |-  F/ z [ y  /  x ] ph
8 sbequ 1812 . . 3  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
96, 7, 8cbvral 2648 . 2  |-  ( A. z  e.  A  [
z  /  x ] ph 
<-> 
A. y  e.  A  [ y  /  x ] ph )
104, 9bitri 183 1  |-  ( A. x  e.  A  ph  <->  A. y  e.  A  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   [wsb 1735   A.wral 2414
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419
This theorem is referenced by:  sbralie  2665  rspsbc  2986  ralxpf  4680
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