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Theorem ecopover 6534
 Description: Assuming that operation is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ecopopr.1
ecopopr.com
ecopopr.cl
ecopopr.ass
ecopopr.can
Assertion
Ref Expression
ecopover
Distinct variable groups:   ,,,,,,   ,,,,,,
Allowed substitution hints:   (,,,,,)

Proof of Theorem ecopover
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . 5
21relopabi 4672 . . . 4
32a1i 9 . . 3
4 ecopopr.com . . . . 5
51, 4ecopovsym 6532 . . . 4
7 ecopopr.cl . . . . 5
8 ecopopr.ass . . . . 5
9 ecopopr.can . . . . 5
101, 4, 7, 8, 9ecopovtrn 6533 . . . 4
12 vex 2692 . . . . . . . . . . 11
13 vex 2692 . . . . . . . . . . 11
1412, 13, 4caovcom 5935 . . . . . . . . . 10
151ecopoveq 6531 . . . . . . . . . 10
1614, 15mpbiri 167 . . . . . . . . 9
1716anidms 395 . . . . . . . 8
1817rgen2a 2489 . . . . . . 7
19 breq12 3941 . . . . . . . . 9
2019anidms 395 . . . . . . . 8
2120ralxp 4689 . . . . . . 7
2218, 21mpbir 145 . . . . . 6
2322rspec 2487 . . . . 5
2423a1i 9 . . . 4
25 opabssxp 4620 . . . . . . 7
261, 25eqsstri 3133 . . . . . 6
2726ssbri 3979 . . . . 5
28 brxp 4577 . . . . . 6
2928simplbi 272 . . . . 5
3027, 29syl 14 . . . 4
3124, 30impbid1 141 . . 3
323, 6, 11, 31iserd 6462 . 2
3332mptru 1341 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1332   wtru 1333  wex 1469   wcel 1481  wral 2417  cop 3534   class class class wbr 3936  copab 3995   cxp 4544   wrel 4551  (class class class)co 5781   wer 6433 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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fv 5138  df-ov 5784  df-er 6436 This theorem is referenced by: (None)
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