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Mirrors > Home > ILE Home > Th. List > subeq0 | Unicode version |
Description: If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.) |
Ref | Expression |
---|---|
subeq0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subid 8117 | . . . 4 | |
2 | 1 | adantl 275 | . . 3 |
3 | 2 | eqeq2d 2177 | . 2 |
4 | subcan2 8123 | . . 3 | |
5 | 4 | 3anidm23 1287 | . 2 |
6 | 3, 5 | bitr3d 189 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1343 wcel 2136 (class class class)co 5842 cc 7751 cc0 7753 cmin 8069 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-setind 4514 ax-resscn 7845 ax-1cn 7846 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-addcom 7853 ax-addass 7855 ax-distr 7857 ax-i2m1 7858 ax-0id 7861 ax-rnegex 7862 ax-cnre 7864 |
This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-sub 8071 |
This theorem is referenced by: subeq0i 8178 subeq0d 8217 subne0d 8218 subeq0ad 8219 cju 8856 fzocongeq 11796 cnmet 13170 |
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