Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7881 | . 2 |
4 | addcl 7713 | . . 3 | |
5 | subadd 7933 | . . 3 | |
6 | 4, 1, 2, 5 | syl3anc 1201 | . 2 |
7 | 3, 6 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 (class class class)co 5742 cc 7586 caddc 7591 cmin 7901 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 |
This theorem is referenced by: pncan2 7937 addsubass 7940 pncan3oi 7946 subid1 7950 nppcan2 7961 pncand 8042 nn1m1nn 8702 nnsub 8723 elnn0nn 8977 zrevaddcl 9062 nzadd 9064 elz2 9080 qrevaddcl 9392 irradd 9394 fzrev3 9822 elfzp1b 9832 fzrevral3 9842 fzval3 9936 subsq2 10355 bcp1nk 10463 bcp1m1 10466 bcpasc 10467 shftlem 10543 shftval5 10556 fsump1 11144 mptfzshft 11166 telfsumo 11190 fsumparts 11194 bcxmas 11213 isum1p 11216 geolim 11235 mertenslem2 11260 mertensabs 11261 eftlub 11310 effsumlt 11312 eirraplem 11395 dvdsadd 11448 prmind2 11713 dvexp 12755 |
Copyright terms: Public domain | W3C validator |