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

Theorem ecopovsymg 6205
 Description: Assuming the operation is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by Jim Kingdon, 1-Sep-2019.)
Hypotheses
Ref Expression
ecopopr.1
ecopoprg.com
Assertion
Ref Expression
ecopovsymg
Distinct variable groups:   ,,,,,,   ,,,,,,
Allowed substitution hints:   (,,,,,)   (,,,,,)   (,,,,,)

Proof of Theorem ecopovsymg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . 5
2 opabssxp 4414 . . . . 5
31, 2eqsstri 2975 . . . 4
43brel 4392 . . 3
5 eqid 2040 . . . 4
6 breq1 3767 . . . . 5
7 breq2 3768 . . . . 5
86, 7bibi12d 224 . . . 4
9 breq2 3768 . . . . 5
10 breq1 3767 . . . . 5
119, 10bibi12d 224 . . . 4
12 ecopoprg.com . . . . . . . . 9
1312adantl 262 . . . . . . . 8
14 simpll 481 . . . . . . . 8
15 simprr 484 . . . . . . . 8
1613, 14, 15caovcomd 5657 . . . . . . 7
17 simplr 482 . . . . . . . 8
18 simprl 483 . . . . . . . 8
1913, 17, 18caovcomd 5657 . . . . . . 7
2016, 19eqeq12d 2054 . . . . . 6
21 eqcom 2042 . . . . . 6
2220, 21syl6bb 185 . . . . 5
231ecopoveq 6201 . . . . 5
241ecopoveq 6201 . . . . . 6
2524ancoms 255 . . . . 5
2622, 23, 253bitr4d 209 . . . 4
275, 8, 11, 262optocl 4417 . . 3
284, 27syl 14 . 2
2928ibi 165 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243  wex 1381   wcel 1393  cop 3378   class class class wbr 3764  copab 3817   cxp 4343  (class class class)co 5512 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-iota 4867  df-fv 4910  df-ov 5515 This theorem is referenced by:  ecopoverg  6207
 Copyright terms: Public domain W3C validator