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Theorem eusvobj2 5441
Description: Specify the same property in two ways when class is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
eusvobj1.1  _V
Assertion
Ref Expression
eusvobj2
Distinct variable groups:   ,,   ,
Allowed substitution hint:   ()

Proof of Theorem eusvobj2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3430 . . 3  {  |  }  { }
2 eleq2 2098 . . . . . 6  {  |  }  { } 
{  |  } 
{ }
3 abid 2025 . . . . . 6  {  |  }
4 elsn 3382 . . . . . 6  { }
52, 3, 43bitr3g 211 . . . . 5  {  |  }  { }
6 nfre1 2359 . . . . . . . . 9  F/
76nfab 2179 . . . . . . . 8  F/_ {  |  }
87nfeq1 2184 . . . . . . 7  F/ {  |  }  { }
9 eusvobj1.1 . . . . . . . . 9  _V
109elabrex 5340 . . . . . . . 8  {  |  }
11 eleq2 2098 . . . . . . . . 9  {  |  }  { }  {  |  }  { }
129elsnc 3390 . . . . . . . . . 10  { }
13 eqcom 2039 . . . . . . . . . 10
1412, 13bitri 173 . . . . . . . . 9  { }
1511, 14syl6bb 185 . . . . . . . 8  {  |  }  { }  {  |  }
1610, 15syl5ib 143 . . . . . . 7  {  |  }  { }
178, 16ralrimi 2384 . . . . . 6  {  |  }  { }
18 eqeq1 2043 . . . . . . 7
1918ralbidv 2320 . . . . . 6
2017, 19syl5ibrcom 146 . . . . 5  {  |  }  { }
215, 20sylbid 139 . . . 4  {  |  }  { }
2221exlimiv 1486 . . 3  {  |  }  { }
231, 22sylbi 114 . 2
24 euex 1927 . . 3
25 rexm 3314 . . . 4
2625exlimiv 1486 . . 3
27 r19.2m 3303 . . . 4
2827ex 108 . . 3
2924, 26, 283syl 17 . 2
3023, 29impbid 120 1
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wceq 1242  wex 1378   wcel 1390  weu 1897   {cab 2023  wral 2300  wrex 2301   _Vcvv 2551   {csn 3367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-sn 3373
This theorem is referenced by:  eusvobj1  5442
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