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Theorem bnj602 31742
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 4293 . . 3  |-  ( X  =  Y  ->  (
y R X  <->  y R Y ) )
21rabbidv 2962 . 2  |-  ( X  =  Y  ->  { y  e.  A  |  y R X }  =  { y  e.  A  |  y R Y } )
3 df-bnj14 31511 . 2  |-  pred ( X ,  A ,  R )  =  {
y  e.  A  | 
y R X }
4 df-bnj14 31511 . 2  |-  pred ( Y ,  A ,  R )  =  {
y  e.  A  | 
y R Y }
52, 3, 43eqtr4g 2498 1  |-  ( X  =  Y  ->  pred ( X ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364   {crab 2717   class class class wbr 4289    predc-bnj14 31510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-bnj14 31511
This theorem is referenced by:  bnj601  31747  bnj852  31748  bnj18eq1  31754  bnj938  31764  bnj1125  31817  bnj1148  31821  bnj1318  31850  bnj1442  31874  bnj1450  31875  bnj1452  31877  bnj1463  31880  bnj1529  31895
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