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Theorem prtlem9 31851
Description: Lemma for prter3 31869. (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 2929 . 2  |-  ( A  e.  B  <->  E. x  e.  B  x  =  A )
2 eceq1 7302 . . 3  |-  ( x  =  A  ->  [ x ]  .~  =  [ A ]  .~  )
32reximi 2869 . 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 1403    e. wcel 1840   E.wrex 2752   [cec 7264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-br 4393  df-opab 4451  df-xp 4946  df-cnv 4948  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-ec 7268
This theorem is referenced by: (None)
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