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Mirrors > Home > ILE Home > Th. List > caov31 | Unicode version |
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) |
Ref | Expression |
---|---|
caov.1 | |
caov.2 | |
caov.3 | |
caov.com | |
caov.ass |
Ref | Expression |
---|---|
caov31 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caov.1 | . . . 4 | |
2 | caov.3 | . . . 4 | |
3 | caov.2 | . . . 4 | |
4 | caov.ass | . . . 4 | |
5 | 1, 2, 3, 4 | caovass 5924 | . . 3 |
6 | caov.com | . . . 4 | |
7 | 1, 2, 3, 6, 4 | caov12 5952 | . . 3 |
8 | 5, 7 | eqtri 2158 | . 2 |
9 | 1, 3, 2, 6, 4 | caov32 5951 | . 2 |
10 | 2, 1, 3, 6, 4 | caov32 5951 | . . 3 |
11 | 2, 1, 3, 4 | caovass 5924 | . . 3 |
12 | 10, 11 | eqtr3i 2160 | . 2 |
13 | 8, 9, 12 | 3eqtr4i 2168 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1331 wcel 1480 cvv 2681 (class class class)co 5767 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 |
This theorem is referenced by: caov13 5954 |
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