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Theorem cbvsetlike 28838
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 28825 . . . 4  |-  ( x  =  z  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A , 
z ) )
21eleq1d 2536 . . 3  |-  ( x  =  z  ->  ( Pred ( R ,  A ,  x )  e.  _V  <->  Pred ( R ,  A ,  z )  e. 
_V ) )
32cbvralv 3088 . 2  |-  ( A. x  e.  A  Pred ( R ,  A ,  x )  e.  _V  <->  A. z  e.  A  Pred ( R ,  A , 
z )  e.  _V )
4 predeq3 28825 . . . 4  |-  ( y  =  z  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A , 
z ) )
54eleq1d 2536 . . 3  |-  ( y  =  z  ->  ( Pred ( R ,  A ,  y )  e. 
_V 
<-> 
Pred ( R ,  A ,  z )  e.  _V ) )
65cbvralv 3088 . 2  |-  ( A. y  e.  A  Pred ( R ,  A , 
y )  e.  _V  <->  A. z  e.  A  Pred ( R ,  A , 
z )  e.  _V )
73, 6bitr4i 252 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 setvar class
Syntax hints:    <-> wb 184    e. wcel 1767   A.wral 2814   _Vcvv 3113   Predcpred 28820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-pred 28821
This theorem is referenced by: (None)
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