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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 8097 | . . 3 | |
2 | 1 | 3ad2ant3 1015 | . 2 |
3 | oveq1 5860 | . . . 4 | |
4 | simpl1 995 | . . . . . . 7 | |
5 | simpl3 997 | . . . . . . 7 | |
6 | simprl 526 | . . . . . . 7 | |
7 | 4, 5, 6 | addassd 7942 | . . . . . 6 |
8 | simprr 527 | . . . . . . 7 | |
9 | 8 | oveq2d 5869 | . . . . . 6 |
10 | addid1 8057 | . . . . . . 7 | |
11 | 4, 10 | syl 14 | . . . . . 6 |
12 | 7, 9, 11 | 3eqtrd 2207 | . . . . 5 |
13 | simpl2 996 | . . . . . . 7 | |
14 | 13, 5, 6 | addassd 7942 | . . . . . 6 |
15 | 8 | oveq2d 5869 | . . . . . 6 |
16 | addid1 8057 | . . . . . . 7 | |
17 | 13, 16 | syl 14 | . . . . . 6 |
18 | 14, 15, 17 | 3eqtrd 2207 | . . . . 5 |
19 | 12, 18 | eqeq12d 2185 | . . . 4 |
20 | 3, 19 | syl5ib 153 | . . 3 |
21 | oveq1 5860 | . . 3 | |
22 | 20, 21 | impbid1 141 | . 2 |
23 | 2, 22 | rexlimddv 2592 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 wrex 2449 (class class class)co 5853 cc 7772 cc0 7774 caddc 7777 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-resscn 7866 ax-1cn 7867 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 |
This theorem is referenced by: addcan2i 8102 addcan2d 8104 muleqadd 8586 |
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