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Theorem caov4d 6002
 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
caovd.4
caovd.cl
Assertion
Ref Expression
caov4d
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,

Proof of Theorem caov4d
StepHypRef Expression
1 caovd.2 . . . 4
2 caovd.3 . . . 4
3 caovd.4 . . . 4
4 caovd.com . . . 4
5 caovd.ass . . . 4
61, 2, 3, 4, 5caov12d 5999 . . 3
76oveq2d 5837 . 2
8 caovd.1 . . 3
9 caovd.cl . . . 4
109, 2, 3caovcld 5971 . . 3
115, 8, 1, 10caovassd 5977 . 2
129, 1, 3caovcld 5971 . . 3
135, 8, 2, 12caovassd 5977 . 2
147, 11, 133eqtr4d 2200 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   w3a 963   wceq 1335   wcel 2128  (class class class)co 5821 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-iota 5134  df-fv 5177  df-ov 5824 This theorem is referenced by:  caov411d  6003  caov42d  6004  ecopovtrn  6574  ecopovtrng  6577  addcmpblnq  7281  mulcmpblnq  7282  ordpipqqs  7288  distrnqg  7301  ltsonq  7312  ltanqg  7314  ltmnqg  7315  addcmpblnq0  7357  mulcmpblnq0  7358  distrnq0  7373  prarloclemlo  7408  addlocprlemeqgt  7446  addcanprleml  7528  recexprlem1ssl  7547  recexprlem1ssu  7548  mulcmpblnrlemg  7654  distrsrg  7673  ltasrg  7684  mulgt0sr  7692  prsradd  7700  axdistr  7788
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