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Mirrors > Home > ILE Home > Th. List > add32 | Unicode version |
Description: Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.) |
Ref | Expression |
---|---|
add32 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcom 7899 | . . . 4 | |
2 | 1 | oveq2d 5790 | . . 3 |
3 | 2 | 3adant1 999 | . 2 |
4 | addass 7750 | . 2 | |
5 | addass 7750 | . . 3 | |
6 | 5 | 3com23 1187 | . 2 |
7 | 3, 4, 6 | 3eqtr4d 2182 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 (class class class)co 5774 cc 7618 caddc 7623 |
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 2121 ax-addcom 7720 ax-addass 7722 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 |
This theorem is referenced by: add32r 7922 add32i 7926 add32d 7930 cnegexlem2 7938 cnegexlem3 7939 2addsub 7976 opeo 11594 |
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