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Mirrors > Home > MPE Home > Th. List > necon3bd | Structured version Visualization version GIF version |
Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
necon3bd.1 | ⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓)) |
Ref | Expression |
---|---|
necon3bd | ⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nne 2942 | . . 3 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵) | |
2 | necon3bd.1 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓)) | |
3 | 1, 2 | biimtrid 241 | . 2 ⊢ (𝜑 → (¬ 𝐴 ≠ 𝐵 → 𝜓)) |
4 | 3 | con1d 145 | 1 ⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1539 ≠ wne 2938 |
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 2939 |
This theorem is referenced by: necon2ad 2953 nssne1 4043 nssne2 4044 disjne 4453 nbrne1 5166 nbrne2 5167 peano5 7886 peano5OLD 7887 oeeui 8604 domdifsn 9056 ac6sfi 9289 inf3lem2 9626 cnfcom3lem 9700 dfac9 10133 fin23lem21 10336 dedekindle 11382 zneo 12649 modirr 13911 sqrmo 15202 reusq0 15413 pc2dvds 16816 pcadd 16826 oddprmdvds 16840 4sqlem11 16892 latnlej 18413 sylow2blem3 19531 irredn0 20314 irredn1 20317 isnzr2 20409 lssvneln0 20706 lspsnne2 20876 lspfixed 20886 lspindpi 20890 lsmcv 20899 lspsolv 20901 coe1tmmul 22019 dfac14 23342 fbdmn0 23558 filufint 23644 flimfnfcls 23752 alexsubALTlem2 23772 evth 24705 cphsqrtcl2 24934 ovolicc2lem4 25269 lhop1lem 25765 lhop1 25766 lhop2 25767 lhop 25768 deg1add 25856 abelthlem2 26180 logcnlem2 26387 angpined 26571 asinneg 26627 dmgmaddn0 26763 lgsne0 27074 lgsqr 27090 lgsquadlem2 27120 lgsquadlem3 27121 axlowdimlem17 28483 spansncvi 31172 zarcmplem 33159 broutsideof2 35398 unblimceq0lem 35685 poimirlem28 36819 dvasin 36875 dvacos 36876 nninfnub 36922 dvrunz 37125 lsatcvatlem 38222 lkrlsp2 38276 opnlen0 38361 2llnne2N 38582 lnnat 38601 llnn0 38690 lplnn0N 38721 lplnllnneN 38730 llncvrlpln2 38731 llncvrlpln 38732 lvoln0N 38765 lplncvrlvol2 38789 lplncvrlvol 38790 dalempnes 38825 dalemqnet 38826 dalemcea 38834 dalem3 38838 cdlema1N 38965 cdlemb 38968 paddasslem5 38998 llnexchb2lem 39042 osumcllem4N 39133 pexmidlem1N 39144 lhp2lt 39175 lhp2atne 39208 lhp2at0ne 39210 4atexlemunv 39240 4atexlemex2 39245 trlne 39359 trlval4 39362 cdlemc4 39368 cdleme11dN 39436 cdleme11h 39440 cdlemednuN 39474 cdleme20j 39492 cdleme20k 39493 cdleme21at 39502 cdleme35f 39628 cdlemg11b 39816 dia2dimlem1 40238 dihmeetlem3N 40479 dihmeetlem15N 40495 dochsnnz 40624 dochexmidlem1 40634 dochexmidlem7 40640 mapdindp3 40896 fltne 41688 pellexlem1 41869 dfac21 42110 pm13.14 43470 uzlidlring 46915 suppdm 47278 |
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