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Theorem bnj1152 29394
Description: Technical lemma for bnj69 29406. 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 4400 . 2  |-  ( y  =  Y  ->  (
y R X  <->  Y R X ) )
2 df-bnj14 29081 . 2  |-  pred ( X ,  A ,  R )  =  {
y  e.  A  | 
y R X }
31, 2elrab2 3211 1  |-  ( Y  e.  pred ( X ,  A ,  R )  <->  ( Y  e.  A  /\  Y R X ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    /\ wa 369    e. wcel 1844   class class class wbr 4397    predc-bnj14 29080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-bnj14 29081
This theorem is referenced by:  bnj1175  29400  bnj1177  29402  bnj1388  29429
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