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Theorem eusvobj2 5529
 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 3469 . . 3
2 eleq2 2143 . . . . . 6
3 abid 2070 . . . . . 6
4 velsn 3423 . . . . . 6
52, 3, 43bitr3g 220 . . . . 5
6 nfre1 2408 . . . . . . . . 9
76nfab 2224 . . . . . . . 8
87nfeq1 2229 . . . . . . 7
9 eusvobj1.1 . . . . . . . . 9
109elabrex 5429 . . . . . . . 8
11 eleq2 2143 . . . . . . . . 9
129elsn 3422 . . . . . . . . . 10
13 eqcom 2084 . . . . . . . . . 10
1412, 13bitri 182 . . . . . . . . 9
1511, 14syl6bb 194 . . . . . . . 8
1610, 15syl5ib 152 . . . . . . 7
178, 16ralrimi 2433 . . . . . 6
18 eqeq1 2088 . . . . . . 7
1918ralbidv 2369 . . . . . 6
2017, 19syl5ibrcom 155 . . . . 5
215, 20sylbid 148 . . . 4
2221exlimiv 1530 . . 3
231, 22sylbi 119 . 2
24 euex 1972 . . 3
25 rexm 3348 . . . 4
2625exlimiv 1530 . . 3
27 r19.2m 3336 . . . 4
2827ex 113 . . 3
2924, 26, 283syl 17 . 2
3023, 29impbid 127 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 103   wceq 1285  wex 1422   wcel 1434  weu 1942  cab 2068  wral 2349  wrex 2350  cvv 2602  csn 3406 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-sn 3412 This theorem is referenced by:  eusvobj1  5530
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