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Mirrors > Home > MPE Home > Th. List > necon3bii | Structured version Visualization version GIF version |
Description: Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.) |
Ref | Expression |
---|---|
necon3bii.1 | ⊢ (𝐴 = 𝐵 ↔ 𝐶 = 𝐷) |
Ref | Expression |
---|---|
necon3bii | ⊢ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon3bii.1 | . . 3 ⊢ (𝐴 = 𝐵 ↔ 𝐶 = 𝐷) | |
2 | 1 | necon3abii 2984 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐶 = 𝐷) |
3 | df-ne 2938 | . 2 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷) | |
4 | 2, 3 | bitr4i 278 | 1 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 = 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: necom 2991 neeq1i 3002 neeq2i 3003 neeq12i 3004 rnsnn0 6229 onoviun 8381 onnseq 8382 intrnfi 9453 wdomtr 9612 noinfep 9697 wemapwe 9734 scott0s 9925 cplem1 9926 karden 9932 acndom2 10091 dfac5lem3 10162 fin23lem31 10380 fin23lem40 10388 isf34lem5 10415 isf34lem7 10416 isf34lem6 10417 axrrecex 11200 negne0bi 11579 rpnnen1lem4 13019 rpnnen1lem5 13020 fseqsupcl 14014 limsupgre 15513 isercolllem3 15699 rpnnen2lem12 16257 ruclem11 16272 3dvds 16364 prmreclem6 16954 0ram 17053 0ram2 17054 0ramcl 17056 gsumval2 18711 ghmrn 19259 gexex 19885 gsumval3 19939 subdrgint 20820 iinopn 22923 cnconn 23445 1stcfb 23468 qtopeu 23739 fbasrn 23907 alexsublem 24067 evth 25004 minveclem1 25471 minveclem3b 25475 ovollb2 25537 ovolunlem1a 25544 ovolunlem1 25545 ovoliunlem1 25550 ovoliun2 25554 ioombl1lem4 25609 uniioombllem1 25629 uniioombllem2 25631 uniioombllem6 25636 mbfsup 25712 mbfinf 25713 mbflimsup 25714 itg1climres 25763 itg2monolem1 25799 itg2mono 25802 itg2i1fseq2 25805 sincos4thpi 26569 nosepnelem 27738 axlowdimlem13 28983 eulerpath 30269 siii 30881 minvecolem1 30902 bcsiALT 31207 h1de2bi 31582 h1de2ctlem 31583 nmlnopgt0i 32025 wrdpmtrlast 33095 dimval 33627 dimvalfi 33628 rge0scvg 33909 umgracycusgr 35138 cusgracyclt3v 35140 erdszelem5 35179 cvmsss2 35258 elrn3 35741 rankeq1o 36152 fin2so 37593 heicant 37641 scottn0f 38156 psspwb 42245 fnwe2lem2 43039 sqrtcval 43630 |
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