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Mirrors > Home > MPE Home > Th. List > neeqtrrd | Structured version Visualization version GIF version |
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
Ref | Expression |
---|---|
neeqtrrd.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
neeqtrrd.2 | ⊢ (𝜑 → 𝐶 = 𝐵) |
Ref | Expression |
---|---|
neeqtrrd | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeqtrrd.1 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | neeqtrrd.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐵) | |
3 | 2 | eqcomd 2740 | . 2 ⊢ (𝜑 → 𝐵 = 𝐶) |
4 | 1, 3 | neeqtrd 3007 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
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 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-cleq 2726 df-ne 2938 |
This theorem is referenced by: 3netr4d 3015 ttukeylem7 10552 modsumfzodifsn 13981 expnprm 16935 symgextf1lem 19452 isabvd 20829 flimclslem 24007 chordthmlem 26889 atandmtan 26977 dchrptlem3 27324 noetasuplem4 27795 opphllem6 28774 nrt2irr 30501 unidifsnne 32561 pmtrcnel 33091 pmtrcnel2 33092 cycpmrn 33145 qsdrnglem2 33503 fedgmul 33658 irngnzply1 33705 irredminply 33721 signstfveq0a 34569 subfacp1lem5 35168 ovoliunnfl 37648 voliunnfl 37650 volsupnfl 37651 cdleme40n 40450 cdleme40w 40452 cdlemg33c 40690 cdlemg33e 40692 trlcocnvat 40706 cdlemh2 40798 cdlemh 40799 cdlemj3 40805 cdlemk24-3 40885 cdlemkfid1N 40903 erng1r 40977 dvalveclem 41007 tendoinvcl 41086 tendolinv 41087 tendorinv 41088 dihatlat 41316 mapdpglem18 41671 mapdpglem22 41675 baerlem5amN 41698 baerlem5bmN 41699 baerlem5abmN 41700 mapdindp1 41702 mapdindp4 41705 hdmap14lem4a 41853 uvcn0 42528 prjspner1 42612 nlimsuc 43430 imo72b2lem2 44156 imo72b2 44161 gpg5nbgrvtx03starlem2 47959 gpg5nbgrvtx13starlem2 47962 islindeps2 48328 |
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