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Theorem bnj1418 28748
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 4157 . 2  |-  ( z  =  y  ->  (
z R x  <->  y R x ) )
2 df-bnj14 28392 . . 3  |-  pred (
x ,  A ,  R )  =  {
z  e.  A  | 
z R x }
32bnj1538 28565 . 2  |-  ( z  e.  pred ( x ,  A ,  R )  ->  z R x )
41, 3vtoclga 2961 1  |-  ( y  e.  pred ( x ,  A ,  R )  ->  y R x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   class class class wbr 4154    predc-bnj14 28391
This theorem is referenced by:  bnj1417  28749  bnj1523  28779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-bnj14 28392
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