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Theorem cbvsetlike 24181
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 24171 . . . 4  |-  ( x  =  z  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A , 
z ) )
21eleq1d 2349 . . 3  |-  ( x  =  z  ->  ( Pred ( R ,  A ,  x )  e.  _V  <->  Pred ( R ,  A ,  z )  e. 
_V ) )
32cbvralv 2764 . 2  |-  ( A. x  e.  A  Pred ( R ,  A ,  x )  e.  _V  <->  A. z  e.  A  Pred ( R ,  A , 
z )  e.  _V )
4 predeq3 24171 . . . 4  |-  ( y  =  z  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A , 
z ) )
54eleq1d 2349 . . 3  |-  ( y  =  z  ->  ( Pred ( R ,  A ,  y )  e. 
_V 
<-> 
Pred ( R ,  A ,  z )  e.  _V ) )
65cbvralv 2764 . 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 1684   A.wral 2543   _Vcvv 2788   Predcpred 24167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-pred 24168
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