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Theorem ecopovsym 7009
 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 4953 . . . . 5
31, 2eqsstri 3380 . . . 4
43brel 4929 . . 3
5 eqid 2438 . . . 4
6 breq1 4218 . . . . 5
7 breq2 4219 . . . . 5
86, 7bibi12d 314 . . . 4
9 breq2 4219 . . . . 5
10 breq1 4218 . . . . 5
119, 10bibi12d 314 . . . 4
121ecopoveq 7008 . . . . . 6
13 vex 2961 . . . . . . . . 9
14 vex 2961 . . . . . . . . 9
15 ecopopr.com . . . . . . . . 9
1613, 14, 15caovcom 6247 . . . . . . . 8
17 vex 2961 . . . . . . . . 9
18 vex 2961 . . . . . . . . 9
1917, 18, 15caovcom 6247 . . . . . . . 8
2016, 19eqeq12i 2451 . . . . . . 7
21 eqcom 2440 . . . . . . 7
2220, 21bitri 242 . . . . . 6
2312, 22syl6bb 254 . . . . 5
241ecopoveq 7008 . . . . . 6
2524ancoms 441 . . . . 5
2623, 25bitr4d 249 . . . 4
275, 8, 11, 262optocl 4956 . . 3
284, 27syl 16 . 2
2928ibi 234 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wex 1551   wceq 1653   wcel 1726  cop 3819   class class class wbr 4215  copab 4268   cxp 4879  (class class class)co 6084 This theorem is referenced by:  ecopover  7011 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-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-xp 4887  df-iota 5421  df-fv 5465  df-ov 6087
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