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Theorem ecopovsym 6518
 Description: Assuming the operation is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
ecopopr.1
ecopopr.com
Assertion
Ref Expression
ecopovsym
Distinct variable groups:   ,,,,,,   ,,,,,,
Allowed substitution hints:   (,,,,,)   (,,,,,)   (,,,,,)

Proof of Theorem ecopovsym
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . 5
2 opabssxp 4608 . . . . 5
31, 2eqsstri 3124 . . . 4
43brel 4586 . . 3
5 eqid 2137 . . . 4
6 breq1 3927 . . . . 5
7 breq2 3928 . . . . 5
86, 7bibi12d 234 . . . 4
9 breq2 3928 . . . . 5
10 breq1 3927 . . . . 5
119, 10bibi12d 234 . . . 4
121ecopoveq 6517 . . . . . 6
13 vex 2684 . . . . . . . . 9
14 vex 2684 . . . . . . . . 9
15 ecopopr.com . . . . . . . . 9
1613, 14, 15caovcom 5921 . . . . . . . 8
17 vex 2684 . . . . . . . . 9
18 vex 2684 . . . . . . . . 9
1917, 18, 15caovcom 5921 . . . . . . . 8
2016, 19eqeq12i 2151 . . . . . . 7
21 eqcom 2139 . . . . . . 7
2220, 21bitri 183 . . . . . 6
2312, 22syl6bb 195 . . . . 5
241ecopoveq 6517 . . . . . 6
2524ancoms 266 . . . . 5
2623, 25bitr4d 190 . . . 4
275, 8, 11, 262optocl 4611 . . 3
284, 27syl 14 . 2
2928ibi 175 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1331  wex 1468   wcel 1480  cop 3525   class class class wbr 3924  copab 3983   cxp 4532  (class class class)co 5767 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-iota 5083  df-fv 5126  df-ov 5770 This theorem is referenced by:  ecopover  6520
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