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Mirrors > Home > ILE Home > Th. List > pncan3 | Unicode version |
Description: Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.) |
Ref | Expression |
---|---|
pncan3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2117 | . 2 | |
2 | simpr 109 | . . 3 | |
3 | simpl 108 | . . 3 | |
4 | subcl 7929 | . . . 4 | |
5 | 4 | ancoms 266 | . . 3 |
6 | subadd 7933 | . . 3 | |
7 | 2, 3, 5, 6 | syl3anc 1201 | . 2 |
8 | 1, 7 | mpbii 147 | 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: npcan 7939 nncan 7959 npncan3 7968 negid 7977 pncan3i 8007 pncan3d 8044 subdi 8115 posdif 8185 fzonmapblen 9932 frecfzen2 10168 bernneq2 10381 hashfz 10535 isumshft 11227 dvdssubr 11466 dvef 12783 sincosq2sgn 12835 sincosq3sgn 12836 sincosq4sgn 12837 |
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