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Theorem prtlem9 26713
Description: Lemma for prter3 26731. (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 2753 . 2  |-  ( A  e.  B  <->  E. x  e.  B  x  =  A )
2 eceq1 6941 . . 3  |-  ( x  =  A  ->  [ x ]  .~  =  [ A ]  .~  )
32reximi 2813 . 2  |-  ( E. x  e.  B  x  =  A  ->  E. x  e.  B  [ x ]  .~  =  [ A ]  .~  )
41, 3sylbi 188 1  |-  ( A  e.  B  ->  E. x  e.  B  [ x ]  .~  =  [ A ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   E.wrex 2706   [cec 6903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-ec 6907
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