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Theorem cbvralsv 2734
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 1539 . . 3  |-  F/ z
ph
2 nfs1v 1951 . . 3  |-  F/ x [ z  /  x ] ph
3 sbequ12 1782 . . 3  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
41, 2, 3cbvral 2714 . 2  |-  ( A. x  e.  A  ph  <->  A. z  e.  A  [ z  /  x ] ph )
5 nfv 1539 . . . 4  |-  F/ y
ph
65nfsb 1958 . . 3  |-  F/ y [ z  /  x ] ph
7 nfv 1539 . . 3  |-  F/ z [ y  /  x ] ph
8 sbequ 1851 . . 3  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
96, 7, 8cbvral 2714 . 2  |-  ( A. z  e.  A  [
z  /  x ] ph 
<-> 
A. y  e.  A  [ y  /  x ] ph )
104, 9bitri 184 1  |-  ( A. x  e.  A  ph  <->  A. y  e.  A  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   [wsb 1773   A.wral 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473
This theorem is referenced by:  sbralie  2736  rspsbc  3060  ralxpf  4788
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