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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 7150 | . . . 4 | |
2 | 1 | oveq2d 5528 | . . 3 |
3 | 2 | 3adant1 922 | . 2 |
4 | addass 7011 | . 2 | |
5 | addass 7011 | . . 3 | |
6 | 5 | 3com23 1110 | . 2 |
7 | 3, 4, 6 | 3eqtr4d 2082 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 (class class class)co 5512 cc 6887 caddc 6892 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-addcom 6984 ax-addass 6986 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: add32r 7171 add32i 7175 add32d 7179 cnegexlem2 7187 cnegexlem3 7188 2addsub 7225 iseqshft2 9232 |
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