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Theorem eusvobj2 6585
 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:   ,,   ,
Allowed substitution hint:   ()

Proof of Theorem eusvobj2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3877 . . 3
2 eleq2 2499 . . . . . 6
3 abid 2426 . . . . . 6
4 elsn 3831 . . . . . 6
52, 3, 43bitr3g 280 . . . . 5
6 nfre1 2764 . . . . . . . . 9
76nfab 2578 . . . . . . . 8
87nfeq1 2583 . . . . . . 7
9 eusvobj1.1 . . . . . . . . 9
109elabrex 5988 . . . . . . . 8
11 eleq2 2499 . . . . . . . . 9
129elsnc 3839 . . . . . . . . . 10
13 eqcom 2440 . . . . . . . . . 10
1412, 13bitri 242 . . . . . . . . 9
1511, 14syl6bb 254 . . . . . . . 8
1610, 15syl5ib 212 . . . . . . 7
178, 16ralrimi 2789 . . . . . 6
18 eqeq1 2444 . . . . . . 7
1918ralbidv 2727 . . . . . 6
2017, 19syl5ibrcom 215 . . . . 5
215, 20sylbid 208 . . . 4
2221exlimiv 1645 . . 3
231, 22sylbi 189 . 2
24 euex 2306 . . 3
25 rexn0 3732 . . . 4
2625exlimiv 1645 . . 3
27 r19.2z 3719 . . . 4
2827ex 425 . . 3
2924, 26, 283syl 19 . 2
3023, 29impbid 185 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178  wex 1551   wceq 1653   wcel 1726  weu 2283  cab 2424   wne 2601  wral 2707  wrex 2708  cvv 2958  c0 3630  csn 3816 This theorem is referenced by:  eusvobj1  6586 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-nul 3631  df-sn 3822
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