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Theorem eusvobj2 6289
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
Assertion
Ref Expression
eusvobj2
Distinct variable groups:   , ,   ,

Proof of Theorem eusvobj2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 euabsn2 4101 . . 3
2 eleq2 2530 . . . . . 6
3 abid 2444 . . . . . 6
4 elsn 4043 . . . . . 6
52, 3, 43bitr3g 287 . . . . 5
6 nfre1 2918 . . . . . . . . 9
76nfab 2623 . . . . . . . 8
87nfeq1 2634 . . . . . . 7
9 eusvobj1.1 . . . . . . . . 9
109elabrex 6155 . . . . . . . 8
11 eleq2 2530 . . . . . . . . 9
129elsnc 4053 . . . . . . . . . 10
13 eqcom 2466 . . . . . . . . . 10
1412, 13bitri 249 . . . . . . . . 9
1511, 14syl6bb 261 . . . . . . . 8
1610, 15syl5ib 219 . . . . . . 7
178, 16ralrimi 2857 . . . . . 6
18 eqeq1 2461 . . . . . . 7
1918ralbidv 2896 . . . . . 6
2017, 19syl5ibrcom 222 . . . . 5
215, 20sylbid 215 . . . 4
2221exlimiv 1722 . . 3
231, 22sylbi 195 . 2
24 euex 2308 . . 3
25 rexn0 3932 . . . 4
2625exlimiv 1722 . . 3
27 r19.2z 3918 . . . 4
2827ex 434 . . 3
2924, 26, 283syl 20 . 2
3023, 29impbid 191 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  E.wex 1612  e.wcel 1818  E!weu 2282  {cab 2442  =/=wne 2652  A.wral 2807  E.wrex 2808   cvv 3109   c0 3784  {csn 4029
This theorem is referenced by:  eusvobj1  6290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-nul 3785  df-sn 4030
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