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Theorem bnj602 29722
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 4424 . . 3  |-  ( X  =  Y  ->  (
y R X  <->  y R Y ) )
21rabbidv 3072 . 2  |-  ( X  =  Y  ->  { y  e.  A  |  y R X }  =  { y  e.  A  |  y R Y } )
3 df-bnj14 29490 . 2  |-  pred ( X ,  A ,  R )  =  {
y  e.  A  | 
y R X }
4 df-bnj14 29490 . 2  |-  pred ( Y ,  A ,  R )  =  {
y  e.  A  | 
y R Y }
52, 3, 43eqtr4g 2488 1  |-  ( X  =  Y  ->  pred ( X ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   {crab 2779   class class class wbr 4420    predc-bnj14 29489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-bnj14 29490
This theorem is referenced by:  bnj601  29727  bnj852  29728  bnj18eq1  29734  bnj938  29744  bnj1125  29797  bnj1148  29801  bnj1318  29830  bnj1442  29854  bnj1450  29855  bnj1452  29857  bnj1463  29860  bnj1529  29875
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