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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 2962 | . 2 ⊢ (𝐶 ≠ 𝐷 → ¬ 𝐴 = 𝐵) |
3 | 2 | neqned 2944 | 1 ⊢ (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ≠ wne 2937 |
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 207 df-ne 2938 |
This theorem is referenced by: difn0 4372 imadisjlnd 6100 xpnz 6180 unixp 6303 inf3lem2 9666 infeq5 9674 cantnflem1 9726 iunfictbso 10151 rankcf 10814 hashfun 14472 hashge3el3dif 14522 abssubne0 15351 expnprm 16935 grpn0 19001 pmtr3ncomlem2 19506 pgpfaclem2 20116 isdrng2 20759 gzrngunit 21468 zringunit 21494 prmirredlem 21500 uvcf1 21829 lindfrn 21858 mpfrcl 22126 ply1frcl 22337 dfac14lem 23640 flimclslem 24007 lebnumlem3 25008 pmltpclem2 25497 i1fmullem 25742 fta1glem1 26221 fta1blem 26224 dgrcolem1 26327 plydivlem4 26352 plyrem 26361 facth 26362 fta1lem 26363 vieta1lem1 26366 vieta1lem2 26367 vieta1 26368 aalioulem2 26389 geolim3 26395 logcj 26662 argregt0 26666 argimgt0 26668 argimlt0 26669 logneg2 26671 tanarg 26675 logtayl 26716 cxpsqrt 26759 cxpcn3lem 26804 cxpcn3 26805 dcubic2 26901 dcubic 26903 cubic 26906 asinlem 26925 atandmcj 26966 atancj 26967 atanlogsublem 26972 bndatandm 26986 birthdaylem1 27008 basellem4 27141 dchrn0 27308 lgsne0 27393 usgr2trlncl 29792 nmlno0lem 30821 nmlnop0iALT 32023 eldmne0 32644 preimane 32686 prmidl0 33457 constrrtll 33736 cntnevol 34208 signsvtn0 34563 signstfveq0a 34569 signstfveq0 34570 nepss 35697 elima4 35756 heicant 37641 totbndbnd 37775 cdleme3c 40212 cdleme7e 40229 sn-1ne2 42278 sn-0ne2 42412 uvcn0 42528 compne 44436 stoweidlem39 45994 rrx2vlinest 48590 rrx2linesl 48592 elfvne0 48678 |
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