Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  eusvobj2 Unicode version

Theorem eusvobj2 5800
 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 3624 . . 3
2 eleq2 2218 . . . . . 6
3 abid 2142 . . . . . 6
4 velsn 3573 . . . . . 6
52, 3, 43bitr3g 221 . . . . 5
6 nfre1 2497 . . . . . . . . 9
76nfab 2301 . . . . . . . 8
87nfeq1 2306 . . . . . . 7
9 eusvobj1.1 . . . . . . . . 9
109elabrex 5699 . . . . . . . 8
11 eleq2 2218 . . . . . . . . 9
129elsn 3572 . . . . . . . . . 10
13 eqcom 2156 . . . . . . . . . 10
1412, 13bitri 183 . . . . . . . . 9
1511, 14bitrdi 195 . . . . . . . 8
1610, 15syl5ib 153 . . . . . . 7
178, 16ralrimi 2525 . . . . . 6
18 eqeq1 2161 . . . . . . 7
1918ralbidv 2454 . . . . . 6
2017, 19syl5ibrcom 156 . . . . 5
215, 20sylbid 149 . . . 4
2221exlimiv 1575 . . 3
231, 22sylbi 120 . 2
24 euex 2033 . . 3
25 rexm 3489 . . . 4
2625exlimiv 1575 . . 3
27 r19.2m 3476 . . . 4
2827ex 114 . . 3
2924, 26, 283syl 17 . 2
3023, 29impbid 128 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104   wceq 1332  wex 1469  weu 2003   wcel 2125  cab 2140  wral 2432  wrex 2433  cvv 2709  csn 3556 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-sbc 2934  df-csb 3028  df-sn 3562 This theorem is referenced by:  eusvobj1  5801
 Copyright terms: Public domain W3C validator