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Mirrors > Home > ILE Home > Th. List > addcan2 | Unicode version |
Description: Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
addcan2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnegex 8053 | . . 3 | |
2 | 1 | 3ad2ant3 1005 | . 2 |
3 | oveq1 5831 | . . . 4 | |
4 | simpl1 985 | . . . . . . 7 | |
5 | simpl3 987 | . . . . . . 7 | |
6 | simprl 521 | . . . . . . 7 | |
7 | 4, 5, 6 | addassd 7900 | . . . . . 6 |
8 | simprr 522 | . . . . . . 7 | |
9 | 8 | oveq2d 5840 | . . . . . 6 |
10 | addid1 8013 | . . . . . . 7 | |
11 | 4, 10 | syl 14 | . . . . . 6 |
12 | 7, 9, 11 | 3eqtrd 2194 | . . . . 5 |
13 | simpl2 986 | . . . . . . 7 | |
14 | 13, 5, 6 | addassd 7900 | . . . . . 6 |
15 | 8 | oveq2d 5840 | . . . . . 6 |
16 | addid1 8013 | . . . . . . 7 | |
17 | 13, 16 | syl 14 | . . . . . 6 |
18 | 14, 15, 17 | 3eqtrd 2194 | . . . . 5 |
19 | 12, 18 | eqeq12d 2172 | . . . 4 |
20 | 3, 19 | syl5ib 153 | . . 3 |
21 | oveq1 5831 | . . 3 | |
22 | 20, 21 | impbid1 141 | . 2 |
23 | 2, 22 | rexlimddv 2579 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 wrex 2436 (class class class)co 5824 cc 7730 cc0 7732 caddc 7735 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-resscn 7824 ax-1cn 7825 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-iota 5135 df-fv 5178 df-ov 5827 |
This theorem is referenced by: addcan2i 8058 addcan2d 8060 muleqadd 8542 |
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