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Theorem cbvralsv 2788
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 1609 . . 3  |-  F/ z
ph
2 nfs1v 2058 . . 3  |-  F/ x [ z  /  x ] ph
3 sbequ12 1872 . . 3  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
41, 2, 3cbvral 2773 . 2  |-  ( A. x  e.  A  ph  <->  A. z  e.  A  [ z  /  x ] ph )
5 nfv 1609 . . . 4  |-  F/ y
ph
65nfsb 2061 . . 3  |-  F/ y [ z  /  x ] ph
7 nfv 1609 . . 3  |-  F/ z [ y  /  x ] ph
8 sbequ 2013 . . 3  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
96, 7, 8cbvral 2773 . 2  |-  ( A. z  e.  A  [
z  /  x ] ph 
<-> 
A. y  e.  A  [ y  /  x ] ph )
104, 9bitri 240 1  |-  ( A. x  e.  A  ph  <->  A. y  e.  A  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [wsb 1638   A.wral 2556
This theorem is referenced by:  sbralie  2790  rspsbc  3082  tfinds  4666  tfindes  4669  ralxpf  4846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561
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