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Theorem bnj602 33408
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 4457 . . 3  |-  ( X  =  Y  ->  (
y R X  <->  y R Y ) )
21rabbidv 3110 . 2  |-  ( X  =  Y  ->  { y  e.  A  |  y R X }  =  { y  e.  A  |  y R Y } )
3 df-bnj14 33177 . 2  |-  pred ( X ,  A ,  R )  =  {
y  e.  A  | 
y R X }
4 df-bnj14 33177 . 2  |-  pred ( Y ,  A ,  R )  =  {
y  e.  A  | 
y R Y }
52, 3, 43eqtr4g 2533 1  |-  ( X  =  Y  ->  pred ( X ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   {crab 2821   class class class wbr 4453    predc-bnj14 33176
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 2822  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-bnj14 33177
This theorem is referenced by:  bnj601  33413  bnj852  33414  bnj18eq1  33420  bnj938  33430  bnj1125  33483  bnj1148  33487  bnj1318  33516  bnj1442  33540  bnj1450  33541  bnj1452  33543  bnj1463  33546  bnj1529  33561
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