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Mirrors > Home > MPE Home > Th. List > necon3i | Structured version Visualization version GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.) (Proof shortened by Wolf Lammen, 22-Nov-2019.) |
Ref | Expression |
---|---|
necon3i.1 | ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
necon3i | ⊢ (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon3i.1 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷) | |
2 | 1 | necon3ai 2965 | . 2 ⊢ (𝐶 ≠ 𝐷 → ¬ 𝐴 = 𝐵) |
3 | 2 | neqned 2947 | 1 ⊢ (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ≠ wne 2940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-ne 2941 |
This theorem is referenced by: difn0 4364 xpnz 6158 unixp 6281 inf3lem2 9626 infeq5 9634 cantnflem1 9686 iunfictbso 10111 rankcf 10774 hashfun 14401 hashge3el3dif 14451 abssubne0 15267 expnprm 16839 grpn0 18892 pmtr3ncomlem2 19383 pgpfaclem2 19993 isdrng2 20514 gzrngunit 21211 zringunit 21237 prmirredlem 21243 uvcf1 21566 lindfrn 21595 mpfrcl 21867 ply1frcl 22057 dfac14lem 23341 flimclslem 23708 lebnumlem3 24703 pmltpclem2 25190 i1fmullem 25435 fta1glem1 25907 fta1blem 25910 dgrcolem1 26011 plydivlem4 26033 plyrem 26042 facth 26043 fta1lem 26044 vieta1lem1 26047 vieta1lem2 26048 vieta1 26049 aalioulem2 26070 geolim3 26076 logcj 26338 argregt0 26342 argimgt0 26344 argimlt0 26345 logneg2 26347 tanarg 26351 logtayl 26392 cxpsqrt 26435 cxpcn3lem 26479 cxpcn3 26480 dcubic2 26573 dcubic 26575 cubic 26578 asinlem 26597 atandmcj 26638 atancj 26639 atanlogsublem 26644 bndatandm 26658 birthdaylem1 26680 basellem4 26812 dchrn0 26977 lgsne0 27062 usgr2trlncl 29272 nmlno0lem 30301 nmlnop0iALT 31503 eldmne0 32107 preimane 32150 prmidl0 32831 cntnevol 33512 signsvtn0 33867 signstfveq0a 33873 signstfveq0 33874 nepss 34979 elima4 35039 heicant 36826 totbndbnd 36960 cdleme3c 39404 cdleme7e 39421 uvcn0 41414 sn-1ne2 41481 sn-0ne2 41581 imadisjlnd 43214 compne 43502 stoweidlem39 45054 rrx2vlinest 47515 rrx2linesl 47517 elfvne0 47603 |
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