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Theorem bnj1152 29028
Description: Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1152  |-  ( Y  e.  pred ( X ,  A ,  R )  <->  ( Y  e.  A  /\  Y R X ) )

Proof of Theorem bnj1152
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 breq1 4026 . 2  |-  ( y  =  Y  ->  (
y R X  <->  Y R X ) )
2 df-bnj14 28714 . 2  |-  pred ( X ,  A ,  R )  =  {
y  e.  A  | 
y R X }
31, 2elrab2 2925 1  |-  ( Y  e.  pred ( X ,  A ,  R )  <->  ( Y  e.  A  /\  Y R X ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   class class class wbr 4023    predc-bnj14 28713
This theorem is referenced by:  bnj1175  29034  bnj1177  29036  bnj1388  29063
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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