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Theorem bnj1152 33010
Description: Technical lemma for bnj69 33022. 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 4445 . 2  |-  ( y  =  Y  ->  (
y R X  <->  Y R X ) )
2 df-bnj14 32698 . 2  |-  pred ( X ,  A ,  R )  =  {
y  e.  A  | 
y R X }
31, 2elrab2 3258 1  |-  ( Y  e.  pred ( X ,  A ,  R )  <->  ( Y  e.  A  /\  Y R X ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1762   class class class wbr 4442    predc-bnj14 32697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-bnj14 32698
This theorem is referenced by:  bnj1175  33016  bnj1177  33018  bnj1388  33045
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