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Theorem caov12d 5682
 Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovd.1
caovd.2
caovd.3
caovd.com
caovd.ass
Assertion
Ref Expression
caov12d
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,

Proof of Theorem caov12d
StepHypRef Expression
1 caovd.com . . . 4
2 caovd.1 . . . 4
3 caovd.2 . . . 4
41, 2, 3caovcomd 5657 . . 3
54oveq1d 5527 . 2
6 caovd.ass . . 3
7 caovd.3 . . 3
86, 2, 3, 7caovassd 5660 . 2
96, 3, 2, 7caovassd 5660 . 2
105, 8, 93eqtr3d 2080 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   w3a 885   wceq 1243   wcel 1393  (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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  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-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515 This theorem is referenced by:  caov4d  5685  caovimo  5694  ltaddnq  6505  ltexnqq  6506  enq0tr  6532  mullocprlem  6668  1idprl  6688  1idpru  6689  cauappcvgprlemdisj  6749  mulcmpblnrlemg  6825  lttrsr  6847  ltsosr  6849  0idsr  6852  1idsr  6853  recexgt0sr  6858  mulgt0sr  6862  axmulass  6947
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