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Theorem bnj1418 33576
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 4456 . 2  |-  ( z  =  y  ->  (
z R x  <->  y R x ) )
2 df-bnj14 33222 . . 3  |-  pred (
x ,  A ,  R )  =  {
z  e.  A  | 
z R x }
32bnj1538 33393 . 2  |-  ( z  e.  pred ( x ,  A ,  R )  ->  z R x )
41, 3vtoclga 3182 1  |-  ( y  e.  pred ( x ,  A ,  R )  ->  y R x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   class class class wbr 4453    predc-bnj14 33221
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-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 33222
This theorem is referenced by:  bnj1417  33577  bnj1523  33607
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