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Mirrors > Home > MPE Home > Th. List > 3anbi2d | Structured version Visualization version GIF version |
Description: Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
Ref | Expression |
---|---|
3anbi1d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
3anbi2d | ⊢ (𝜑 → ((𝜃 ∧ 𝜓 ∧ 𝜏) ↔ (𝜃 ∧ 𝜒 ∧ 𝜏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 262 | . 2 ⊢ (𝜑 → (𝜃 ↔ 𝜃)) | |
2 | 3anbi1d.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | 3anbi12d 1438 | 1 ⊢ (𝜑 → ((𝜃 ∧ 𝜓 ∧ 𝜏) ↔ (𝜃 ∧ 𝜒 ∧ 𝜏))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1088 |
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-an 398 df-3an 1090 |
This theorem is referenced by: vtocl3gaf 3569 vtocl3ga 3570 offval22 8074 ereq2 8711 brttrcl 9708 ttrclss 9715 ttrclselem2 9721 wrdl3s3 14913 mhmlem 18945 isdrngrd 20391 isdrngrdOLD 20393 lmodlema 20476 mdetunilem9 22122 neiptoptop 22635 neiptopnei 22636 hausnei 22832 regr1lem2 23244 ustuqtop4 23749 utopsnneiplem 23752 axtg5seg 27716 axtgupdim2 27722 axtgeucl 27723 brbtwn 28157 axlowdim 28219 axeuclidlem 28220 incistruhgr 28339 issubgr2 28529 wwlksnwwlksnon 29169 upgr4cycl4dv4e 29438 isnvlem 29863 csmdsymi 31587 br8d 31839 slmdlema 32348 carsgmon 33313 sitgclg 33341 tgoldbachgt 33675 axtgupdim2ALTV 33680 bnj852 33932 bnj18eq1 33938 bnj938 33948 bnj983 33962 bnj1318 34036 bnj1326 34037 cvmlift3lem4 34313 cvmlift3 34319 br8 34726 br6 34727 br4 34728 brcolinear2 35030 colineardim1 35033 brfs 35051 fscgr 35052 btwnconn1lem7 35065 brsegle 35080 unblimceq0 35383 sdclem2 36610 sdclem1 36611 sdc 36612 fdc 36613 cdleme18d 39166 cdlemk35s 39808 cdlemk39s 39810 monotoddzz 41682 jm2.27 41747 mendlmod 41935 minregex2 42286 fiiuncl 43752 wessf1ornlem 43882 fmulcl 44297 fmuldfeqlem1 44298 fprodcncf 44616 dvmptfprodlem 44660 dvmptfprod 44661 dvnprodlem2 44663 stoweidlem6 44722 stoweidlem8 44724 stoweidlem31 44747 stoweidlem34 44750 stoweidlem43 44759 stoweidlem52 44768 fourierdlem41 44864 fourierdlem48 44870 fourierdlem49 44871 ovnsubaddlem1 45286 ichexmpl2 46138 9gbo 46442 11gbo 46443 lmod1 47173 |
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