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Theorem prtlem9 29177
Description: Lemma for prter3 29195. (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 2885 . 2  |-  ( A  e.  B  <->  E. x  e.  B  x  =  A )
2 eceq1 7250 . . 3  |-  ( x  =  A  ->  [ x ]  .~  =  [ A ]  .~  )
32reximi 2929 . 2  |-  ( E. x  e.  B  x  =  A  ->  E. x  e.  B  [ x ]  .~  =  [ A ]  .~  )
41, 3sylbi 195 1  |-  ( A  e.  B  ->  E. x  e.  B  [ x ]  .~  =  [ A ]  .~  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   E.wrex 2800   [cec 7212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957  df-cnv 4959  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-ec 7216
This theorem is referenced by: (None)
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