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Theorem prtlem9 26732
Description: Lemma for prter3 26750. (Contributed by Rodolfo Medina, 25-Sep-2010.)
Assertion
Ref Expression
prtlem9  |-  ( A  e.  B  ->  E. x  e.  B  [ x ]  .~  =  [ A ]  .~  )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    .~ ( x)

Proof of Theorem prtlem9
StepHypRef Expression
1 risset 2590 . 2  |-  ( A  e.  B  <->  E. x  e.  B  x  =  A )
2 eceq1 6696 . . 3  |-  ( x  =  A  ->  [ x ]  .~  =  [ A ]  .~  )
32reximi 2650 . 2  |-  ( E. x  e.  B  x  =  A  ->  E. x  e.  B  [ x ]  .~  =  [ A ]  .~  )
41, 3sylbi 187 1  |-  ( A  e.  B  ->  E. x  e.  B  [ x ]  .~  =  [ A ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544   [cec 6658
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-ral 2548  df-rex 2549  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-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662
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