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Theorem bnj1418 29070
Description: Property of  pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1418  |-  ( y  e.  pred ( x ,  A ,  R )  ->  y R x )

Proof of Theorem bnj1418
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 breq1 4026 . 2  |-  ( z  =  y  ->  (
z R x  <->  y R x ) )
2 df-bnj14 28714 . . 3  |-  pred (
x ,  A ,  R )  =  {
z  e.  A  | 
z R x }
32bnj1538 28887 . 2  |-  ( z  e.  pred ( x ,  A ,  R )  ->  z R x )
41, 3vtoclga 2849 1  |-  ( y  e.  pred ( x ,  A ,  R )  ->  y R x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   class class class wbr 4023    predc-bnj14 28713
This theorem is referenced by:  bnj1417  29071  bnj1523  29101
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-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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