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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 108 | . . 3 | |
2 | simpl 107 | . . 3 | |
3 | 1, 2 | addcomd 7259 | . 2 |
4 | addcl 7098 | . . 3 | |
5 | subadd 7311 | . . 3 | |
6 | 4, 1, 2, 5 | syl3anc 1169 | . 2 |
7 | 3, 6 | mpbird 165 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 (class class class)co 5532 cc 6979 caddc 6984 cmin 7279 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 |
This theorem is referenced by: pncan2 7315 addsubass 7318 pncan3oi 7324 subid1 7328 nppcan2 7339 pncand 7420 nn1m1nn 8057 nnsub 8077 elnn0nn 8330 zrevaddcl 8401 nzadd 8403 elz2 8419 qrevaddcl 8729 irradd 8731 fzrev3 9104 elfzp1b 9114 fzrevral3 9124 fzval3 9213 subsq2 9582 bcp1nk 9689 bcp1m1 9692 bcpasc 9693 shftlem 9704 shftval5 9717 dvdsadd 10238 prmind2 10502 |
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