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Mirrors > Home > ILE Home > Th. List > pncan | Unicode version |
Description: Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
pncan |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . 3 | |
2 | simpl 108 | . . 3 | |
3 | 1, 2 | addcomd 8049 | . 2 |
4 | addcl 7878 | . . 3 | |
5 | subadd 8101 | . . 3 | |
6 | 4, 1, 2, 5 | syl3anc 1228 | . 2 |
7 | 3, 6 | mpbird 166 | 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 caddc 7756 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: pncan2 8105 addsubass 8108 pncan3oi 8114 subid1 8118 nppcan2 8129 pncand 8210 nn1m1nn 8875 nnsub 8896 elnn0nn 9156 zrevaddcl 9241 nzadd 9243 elz2 9262 qrevaddcl 9582 irradd 9584 fzrev3 10022 elfzp1b 10032 fzrevral3 10042 fzval3 10139 subsq2 10562 bcp1nk 10675 bcp1m1 10678 bcpasc 10679 shftlem 10758 shftval5 10771 fsump1 11361 mptfzshft 11383 telfsumo 11407 fsumparts 11411 bcxmas 11430 isum1p 11433 geolim 11452 mertenslem2 11477 mertensabs 11478 eftlub 11631 effsumlt 11633 eirraplem 11717 dvdsadd 11776 prmind2 12052 fldivp1 12278 prmpwdvds 12285 pockthlem 12286 dvexp 13315 abssinper 13407 lgsvalmod 13560 2sqlem10 13601 |
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