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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 261 | . 2 ⊢ (𝜑 → (𝜃 ↔ 𝜃)) | |
2 | 3anbi1d.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | 3anbi12d 1437 | 1 ⊢ (𝜑 → ((𝜃 ∧ 𝜓 ∧ 𝜏) ↔ (𝜃 ∧ 𝜒 ∧ 𝜏))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1087 |
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 397 df-3an 1089 |
This theorem is referenced by: vtocl3gaf 3568 vtocl3ga 3569 offval22 8076 ereq2 8713 brttrcl 9710 ttrclss 9717 ttrclselem2 9723 wrdl3s3 14917 mhmlem 18981 isdrngrd 20534 isdrngrdOLD 20536 lmodlema 20619 mdetunilem9 22342 neiptoptop 22855 neiptopnei 22856 hausnei 23052 regr1lem2 23464 ustuqtop4 23969 utopsnneiplem 23972 axtg5seg 27971 axtgupdim2 27977 axtgeucl 27978 brbtwn 28412 axlowdim 28474 axeuclidlem 28475 incistruhgr 28594 issubgr2 28784 wwlksnwwlksnon 29424 upgr4cycl4dv4e 29693 isnvlem 30118 csmdsymi 31842 br8d 32094 slmdlema 32606 carsgmon 33599 sitgclg 33627 tgoldbachgt 33961 axtgupdim2ALTV 33966 bnj852 34218 bnj18eq1 34224 bnj938 34234 bnj983 34248 bnj1318 34322 bnj1326 34323 cvmlift3lem4 34599 cvmlift3 34605 br8 35018 br6 35019 br4 35020 brcolinear2 35322 colineardim1 35325 brfs 35343 fscgr 35344 btwnconn1lem7 35357 brsegle 35372 unblimceq0 35686 sdclem2 36913 sdclem1 36914 sdc 36915 fdc 36916 cdleme18d 39469 cdlemk35s 40111 cdlemk39s 40113 monotoddzz 41984 jm2.27 42049 mendlmod 42237 minregex2 42588 fiiuncl 44054 wessf1ornlem 44183 fmulcl 44596 fmuldfeqlem1 44597 fprodcncf 44915 dvmptfprodlem 44959 dvmptfprod 44960 dvnprodlem2 44962 stoweidlem6 45021 stoweidlem8 45023 stoweidlem31 45046 stoweidlem34 45049 stoweidlem43 45058 stoweidlem52 45067 fourierdlem41 45163 fourierdlem48 45169 fourierdlem49 45170 ovnsubaddlem1 45585 ichexmpl2 46437 9gbo 46741 11gbo 46742 lmod1 47261 |
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