MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caov12 Unicode version

Theorem caov12 6424
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
Hypotheses
Ref Expression
caov.1
caov.2
caov.3
caov.com
caov.ass
Assertion
Ref Expression
caov12
Distinct variable groups:   , , ,   , , ,   , , ,   , , ,

Proof of Theorem caov12
StepHypRef Expression
1 caov.1 . . . 4
2 caov.2 . . . 4
3 caov.com . . . 4
41, 2, 3caovcom 6393 . . 3
54oveq1i 6232 . 2
6 caov.3 . . 3
7 caov.ass . . 3
81, 2, 6, 7caovass 6396 . 2
92, 1, 6, 7caovass 6396 . 2
105, 8, 93eqtr3i 2491 1
Colors of variables: wff setvar class
Syntax hints:  =wceq 1370  e.wcel 1758   cvv 3081  (class class class)co 6222
This theorem is referenced by:  caov31  6425  caov4  6427  caovmo  6433  distrnq  9267  ltaddnq  9280  ltexnq  9281  1idpr  9335  prlem934  9339  prlem936  9353  mulcmpblnrlem  9377  ltsosr  9398  0idsr  9401  1idsr  9402  recexsrlem  9407  mulgt0sr  9409  axmulass  9461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3083  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3752  df-if 3906  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4209  df-br 4410  df-iota 5500  df-fv 5545  df-ov 6225
  Copyright terms: Public domain W3C validator