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Theorem bnj602 29263
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 4043 . . 3  |-  ( X  =  Y  ->  (
y R X  <->  y R Y ) )
21rabbidv 2793 . 2  |-  ( X  =  Y  ->  { y  e.  A  |  y R X }  =  { y  e.  A  |  y R Y } )
3 df-bnj14 29030 . 2  |-  pred ( X ,  A ,  R )  =  {
y  e.  A  | 
y R X }
4 df-bnj14 29030 . 2  |-  pred ( Y ,  A ,  R )  =  {
y  e.  A  | 
y R Y }
52, 3, 43eqtr4g 2353 1  |-  ( X  =  Y  ->  pred ( X ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   {crab 2560   class class class wbr 4039    predc-bnj14 29029
This theorem is referenced by:  bnj601  29268  bnj852  29269  bnj18eq1  29275  bnj938  29285  bnj1125  29338  bnj1148  29342  bnj1318  29371  bnj1442  29395  bnj1450  29396  bnj1452  29398  bnj1463  29401  bnj1529  29416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-bnj14 29030
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