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| Mirrors > Home > MPE Home > Th. List > neeqtrd | Structured version Visualization version GIF version | ||
| Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
| Ref | Expression |
|---|---|
| neeqtrd.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
| neeqtrd.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| neeqtrd | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | neeqtrd.1 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
| 2 | neeqtrd.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
| 3 | 2 | neeq2d 3001 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶)) |
| 4 | 1, 3 | mpbid 232 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ≠ wne 2940 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ne 2941 |
| This theorem is referenced by: neeqtrrd 3015 3netr3d 3017 xaddass2 13292 xov1plusxeqvd 13538 smndex2dnrinv 18928 ablsimpgfindlem1 20127 issubdrg 20781 qsssubdrg 21444 ply1scln0 22295 alexsublem 24052 cphsubrglem 25211 cphreccllem 25212 mdegldg 26105 nosep2o 27727 noetainflem4 27785 tglinethru 28644 footexALT 28726 footexlem2 28728 nrt2irr 30492 sdrgdvcl 33301 sdrginvcl 33302 0ringprmidl 33477 0ringmon1p 33583 irngnzply1lem 33740 irngnminplynz 33755 minplym1p 33756 algextdeglem4 33761 poimirlem26 37653 lkrpssN 39164 lnatexN 39781 lhpexle2lem 40011 lhpexle3lem 40013 cdlemg47 40738 cdlemk54 40960 tendoinvcl 41106 lcdlkreqN 41624 mapdh8ab 41779 aks6d1c5lem2 42139 aks6d1c7 42185 jm2.26lem3 43013 stoweidlem36 46051 addmodne 47346 p1modne 47349 m1modne 47350 minusmod5ne 47351 gpg5nbgrvtx03starlem2 48025 gpg5nbgrvtx13starlem2 48028 |
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