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Theorem ecopovsym 6965
 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 4909 . . . . 5
31, 2eqsstri 3338 . . . 4
43brel 4885 . . 3
5 eqid 2404 . . . 4
6 breq1 4175 . . . . 5
7 breq2 4176 . . . . 5
86, 7bibi12d 313 . . . 4
9 breq2 4176 . . . . 5
10 breq1 4175 . . . . 5
119, 10bibi12d 313 . . . 4
121ecopoveq 6964 . . . . . 6
13 vex 2919 . . . . . . . . 9
14 vex 2919 . . . . . . . . 9
15 ecopopr.com . . . . . . . . 9
1613, 14, 15caovcom 6203 . . . . . . . 8
17 vex 2919 . . . . . . . . 9
18 vex 2919 . . . . . . . . 9
1917, 18, 15caovcom 6203 . . . . . . . 8
2016, 19eqeq12i 2417 . . . . . . 7
21 eqcom 2406 . . . . . . 7
2220, 21bitri 241 . . . . . 6
2312, 22syl6bb 253 . . . . 5
241ecopoveq 6964 . . . . . 6
2524ancoms 440 . . . . 5
2623, 25bitr4d 248 . . . 4
275, 8, 11, 262optocl 4912 . . 3
284, 27syl 16 . 2
2928ibi 233 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1547   wceq 1649   wcel 1721  cop 3777   class class class wbr 4172  copab 4225   cxp 4835  (class class class)co 6040 This theorem is referenced by:  ecopover  6967 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-iota 5377  df-fv 5421  df-ov 6043
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