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Mirrors > Home > ILE Home > Th. List > npcan | Unicode version |
Description: Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
npcan |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcl 8154 |
. . 3
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2 | simpr 110 |
. . 3
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3 | 1, 2 | addcomd 8106 |
. 2
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4 | pncan3 8163 |
. . 3
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5 | 4 | ancoms 268 |
. 2
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6 | 3, 5 | eqtrd 2210 |
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-in1 614 ax-in2 615 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-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-setind 4536 ax-resscn 7902 ax-1cn 7903 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-sub 8128 |
This theorem is referenced by: addsubass 8165 npncan 8176 nppcan 8177 nnpcan 8178 subcan2 8180 nnncan 8190 npcand 8270 nn1suc 8936 zlem1lt 9307 zltlem1 9308 peano5uzti 9359 nummac 9426 uzp1 9559 peano2uzr 9583 fz01en 10050 fzsuc2 10076 fseq1m1p1 10092 fzoss2 10169 fzoaddel2 10190 fzosplitsnm1 10206 fzosplitprm1 10231 modfzo0difsn 10392 seq3m1 10465 monoord2 10474 ser3mono 10475 expm1t 10545 expubnd 10574 bcm1k 10735 bcn2 10739 hashfzo 10797 seq3coll 10817 shftlem 10820 shftfvalg 10822 shftfval 10825 iserex 11342 serf0 11355 fsumm1 11419 mptfzshft 11445 binomlem 11486 binom1dif 11490 isumsplit 11494 dvdssub2 11837 |
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