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Mirrors > Home > MPE Home > Th. List > neeq2 | Structured version Visualization version GIF version |
Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.) |
Ref | Expression |
---|---|
neeq2 | ⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
2 | 1 | neeq2d 3002 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ≠ wne 2941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2725 df-ne 2942 |
This theorem is referenced by: psseq2 4089 prproe 4907 fprg 7153 f1dom3el3dif 7268 f1prex 7282 xpord2lem 8128 xpord3lem 8135 dfac5 10123 kmlem4 10148 kmlem14 10158 1re 11214 hashge2el2difr 14442 hashdmpropge2 14444 dvdsle 16253 sgrp2rid2ex 18808 isirred 20233 isnzr2 20297 dmatelnd 21998 mdetdiaglem 22100 mdetunilem1 22114 mdetunilem2 22115 maducoeval2 22142 hausnei 22832 regr1lem2 23244 xrhmeo 24462 axtg5seg 27716 axtgupdim2 27722 axtgeucl 27723 ishlg 27853 tglnpt2 27892 axlowdim1 28217 umgrvad2edg 28470 2pthdlem1 29184 3pthdlem1 29417 upgr3v3e3cycl 29433 upgr4cycl4dv4e 29438 eupth2lem3lem4 29484 3cyclfrgrrn1 29538 4cycl2vnunb 29543 numclwwlkovh 29626 numclwwlkovq 29627 numclwwlk2lem1 29629 numclwlk2lem2f 29630 superpos 31607 signswch 33572 axtgupdim2ALTV 33680 dfrdg4 34923 fvray 35113 linedegen 35115 fvline 35116 linethru 35125 hilbert1.1 35126 knoppndvlem21 35408 poimirlem1 36489 hlsuprexch 38252 3dim1lem5 38337 llni2 38383 lplni2 38408 2llnjN 38438 lvoli2 38452 2lplnj 38491 islinei 38611 cdleme40n 39339 cdlemg33b 39578 ax6e2ndeq 43320 ax6e2ndeqVD 43670 ax6e2ndeqALT 43692 refsum2cnlem1 43721 stoweidlem43 44759 nnfoctbdjlem 45171 elprneb 45739 ichnreuop 46140 inlinecirc02plem 47472 |
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