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Theorem cbvsetlike 24739
Description: Change the bound variable in the statement stating that  R is set-like. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
cbvsetlike  |-  ( A. x  e.  A  Pred ( R ,  A ,  x )  e.  _V  <->  A. y  e.  A  Pred ( R ,  A , 
y )  e.  _V )
Distinct variable groups:    x, R    y, R    x, A    y, A

Proof of Theorem cbvsetlike
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 predeq3 24729 . . . 4  |-  ( x  =  z  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A , 
z ) )
21eleq1d 2424 . . 3  |-  ( x  =  z  ->  ( Pred ( R ,  A ,  x )  e.  _V  <->  Pred ( R ,  A ,  z )  e. 
_V ) )
32cbvralv 2840 . 2  |-  ( A. x  e.  A  Pred ( R ,  A ,  x )  e.  _V  <->  A. z  e.  A  Pred ( R ,  A , 
z )  e.  _V )
4 predeq3 24729 . . . 4  |-  ( y  =  z  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A , 
z ) )
54eleq1d 2424 . . 3  |-  ( y  =  z  ->  ( Pred ( R ,  A ,  y )  e. 
_V 
<-> 
Pred ( R ,  A ,  z )  e.  _V ) )
65cbvralv 2840 . 2  |-  ( A. y  e.  A  Pred ( R ,  A , 
y )  e.  _V  <->  A. z  e.  A  Pred ( R ,  A , 
z )  e.  _V )
73, 6bitr4i 243 1  |-  ( A. x  e.  A  Pred ( R ,  A ,  x )  e.  _V  <->  A. y  e.  A  Pred ( R ,  A , 
y )  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1710   A.wral 2619   _Vcvv 2864   Predcpred 24725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-xp 4774  df-cnv 4776  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-pred 24726
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