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Mirrors > Home > ILE Home > Th. List > add12 | Unicode version |
Description: Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.) |
Ref | Expression |
---|---|
add12 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcom 7867 | . . . 4 | |
2 | 1 | oveq1d 5757 | . . 3 |
3 | 2 | 3adant3 986 | . 2 |
4 | addass 7718 | . 2 | |
5 | addass 7718 | . . 3 | |
6 | 5 | 3com12 1170 | . 2 |
7 | 3, 4, 6 | 3eqtr3d 2158 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 (class class class)co 5742 cc 7586 caddc 7591 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-addcom 7688 ax-addass 7690 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 |
This theorem is referenced by: add4 7891 add12i 7893 add12d 7897 |
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