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Theorem cbvsetlike 25399
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 25389 . . . 4  |-  ( x  =  z  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A , 
z ) )
21eleq1d 2474 . . 3  |-  ( x  =  z  ->  ( Pred ( R ,  A ,  x )  e.  _V  <->  Pred ( R ,  A ,  z )  e. 
_V ) )
32cbvralv 2896 . 2  |-  ( A. x  e.  A  Pred ( R ,  A ,  x )  e.  _V  <->  A. z  e.  A  Pred ( R ,  A , 
z )  e.  _V )
4 predeq3 25389 . . . 4  |-  ( y  =  z  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A , 
z ) )
54eleq1d 2474 . . 3  |-  ( y  =  z  ->  ( Pred ( R ,  A ,  y )  e. 
_V 
<-> 
Pred ( R ,  A ,  z )  e.  _V ) )
65cbvralv 2896 . 2  |-  ( A. y  e.  A  Pred ( R ,  A , 
y )  e.  _V  <->  A. z  e.  A  Pred ( R ,  A , 
z )  e.  _V )
73, 6bitr4i 244 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 177    e. wcel 1721   A.wral 2670   _Vcvv 2920   Predcpred 25385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-xp 4847  df-cnv 4849  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-pred 25386
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