Theorem caov4 6506

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
caov.4

Assertion

Ref	Expression
caov4

Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,

Proof of Theorem caov4

Step	Hyp	Ref	Expression
1		caov.2	. . . 4
2		caov.3	. . . 4
3		caov.4	. . . 4
4		caov.com	. . . 4
5		caov.ass	. . . 4
6	1, 2, 3, 4, 5	caov12 6503	. . 3
7	6	oveq2i 6307	. 2
8		caov.1	. . 3
9		ovex 6324	. . 3
10	8, 1, 9, 5	caovass 6475	. 2
11		ovex 6324	. . 3
12	8, 2, 11, 5	caovass 6475	. 2
13	7, 10, 12	3eqtr4i 2496	1

Syntax hints:  =wceq 1395  e.wcel 1818  cvv 3109  (class class class)co 6296

This theorem is referenced by:  caov42  6508  ecopovtrn  7433  adderpqlem  9353  mulerpqlem  9354  ltmnq  9371  reclem3pr  9448  mulcmpblnrlem  9468  distrsr  9489  ltasr  9498  mulgt0sr  9503  axdistr  9556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-nul 4581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-iota 5556  df-fv 5601  df-ov 6299