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Theorem bnj602 28947
Description: Equality theorem for the  pred function constant. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj602  |-  ( X  =  Y  ->  pred ( X ,  A ,  R )  =  pred ( Y ,  A ,  R ) )

Proof of Theorem bnj602
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 breq2 4027 . . 3  |-  ( X  =  Y  ->  (
y R X  <->  y R Y ) )
21rabbidv 2780 . 2  |-  ( X  =  Y  ->  { y  e.  A  |  y R X }  =  { y  e.  A  |  y R Y } )
3 df-bnj14 28714 . 2  |-  pred ( X ,  A ,  R )  =  {
y  e.  A  | 
y R X }
4 df-bnj14 28714 . 2  |-  pred ( Y ,  A ,  R )  =  {
y  e.  A  | 
y R Y }
52, 3, 43eqtr4g 2340 1  |-  ( X  =  Y  ->  pred ( X ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   {crab 2547   class class class wbr 4023    predc-bnj14 28713
This theorem is referenced by:  bnj601  28952  bnj852  28953  bnj18eq1  28959  bnj938  28969  bnj1125  29022  bnj1148  29026  bnj1318  29055  bnj1442  29079  bnj1450  29080  bnj1452  29082  bnj1463  29085  bnj1529  29100
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-bnj14 28714
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