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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 2966 | . 2 ⊢ (𝐶 ≠ 𝐷 → ¬ 𝐴 = 𝐵) |
3 | 2 | neqned 2948 | 1 ⊢ (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ≠ wne 2941 |
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 2942 |
This theorem is referenced by: difn0 4365 xpnz 6159 unixp 6282 inf3lem2 9624 infeq5 9632 cantnflem1 9684 iunfictbso 10109 rankcf 10772 hashfun 14397 hashge3el3dif 14447 abssubne0 15263 expnprm 16835 grpn0 18856 pmtr3ncomlem2 19342 pgpfaclem2 19952 isdrng2 20371 gzrngunit 21011 zringunit 21036 prmirredlem 21042 uvcf1 21347 lindfrn 21376 mpfrcl 21648 ply1frcl 21837 dfac14lem 23121 flimclslem 23488 lebnumlem3 24479 pmltpclem2 24966 i1fmullem 25211 fta1glem1 25683 fta1blem 25686 dgrcolem1 25787 plydivlem4 25809 plyrem 25818 facth 25819 fta1lem 25820 vieta1lem1 25823 vieta1lem2 25824 vieta1 25825 aalioulem2 25846 geolim3 25852 logcj 26114 argregt0 26118 argimgt0 26120 argimlt0 26121 logneg2 26123 tanarg 26127 logtayl 26168 cxpsqrt 26211 cxpcn3lem 26255 cxpcn3 26256 dcubic2 26349 dcubic 26351 cubic 26354 asinlem 26373 atandmcj 26414 atancj 26415 atanlogsublem 26420 bndatandm 26434 birthdaylem1 26456 basellem4 26588 dchrn0 26753 lgsne0 26838 usgr2trlncl 29017 nmlno0lem 30046 nmlnop0iALT 31248 eldmne0 31852 preimane 31895 prmidl0 32569 cntnevol 33226 signsvtn0 33581 signstfveq0a 33587 signstfveq0 33588 nepss 34687 elima4 34747 heicant 36523 totbndbnd 36657 cdleme3c 39101 cdleme7e 39118 uvcn0 41112 sn-1ne2 41179 sn-0ne2 41279 imadisjlnd 42912 compne 43200 stoweidlem39 44755 rrx2vlinest 47427 rrx2linesl 47429 elfvne0 47515 |
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