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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnegex 8122 |
. . 3
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2 | 1 | 3ad2ant3 1020 |
. 2
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3 | oveq1 5876 |
. . . 4
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4 | simpl1 1000 |
. . . . . . 7
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5 | simpl3 1002 |
. . . . . . 7
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6 | simprl 529 |
. . . . . . 7
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7 | 4, 5, 6 | addassd 7967 |
. . . . . 6
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8 | simprr 531 |
. . . . . . 7
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9 | 8 | oveq2d 5885 |
. . . . . 6
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10 | addid1 8082 |
. . . . . . 7
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11 | 4, 10 | syl 14 |
. . . . . 6
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12 | 7, 9, 11 | 3eqtrd 2214 |
. . . . 5
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13 | simpl2 1001 |
. . . . . . 7
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14 | 13, 5, 6 | addassd 7967 |
. . . . . 6
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15 | 8 | oveq2d 5885 |
. . . . . 6
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16 | addid1 8082 |
. . . . . . 7
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17 | 13, 16 | syl 14 |
. . . . . 6
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18 | 14, 15, 17 | 3eqtrd 2214 |
. . . . 5
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19 | 12, 18 | eqeq12d 2192 |
. . . 4
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20 | 3, 19 | imbitrid 154 |
. . 3
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21 | oveq1 5876 |
. . 3
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22 | 20, 21 | impbid1 142 |
. 2
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23 | 2, 22 | rexlimddv 2599 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-resscn 7891 ax-1cn 7892 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-addass 7901 ax-distr 7903 ax-i2m1 7904 ax-0id 7907 ax-rnegex 7908 ax-cnre 7910 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-iota 5174 df-fv 5220 df-ov 5872 |
This theorem is referenced by: addcan2i 8127 addcan2d 8129 muleqadd 8611 |
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