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Mirrors > Home > MPE Home > Th. List > 3bitr4rd | Structured version Visualization version GIF version |
Description: Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.) |
Ref | Expression |
---|---|
3bitr4d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
3bitr4d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜓)) |
3bitr4d.3 | ⊢ (𝜑 → (𝜏 ↔ 𝜒)) |
Ref | Expression |
---|---|
3bitr4rd | ⊢ (𝜑 → (𝜏 ↔ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3bitr4d.3 | . . 3 ⊢ (𝜑 → (𝜏 ↔ 𝜒)) | |
2 | 3bitr4d.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | bitr4d 270 | . 2 ⊢ (𝜑 → (𝜏 ↔ 𝜓)) |
4 | 3bitr4d.2 | . 2 ⊢ (𝜑 → (𝜃 ↔ 𝜓)) | |
5 | 3, 4 | bitr4d 270 | 1 ⊢ (𝜑 → (𝜏 ↔ 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 |
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 196 |
This theorem is referenced by: inimasn 5469 isof1oidb 6474 oacan 7515 ecdmn0 7676 wemapwe 8477 r1pw 8591 adderpqlem 9655 mulerpqlem 9656 lterpq 9671 ltanq 9672 genpass 9710 readdcan 10089 lemuldiv 10782 msq11 10803 avglt2 11148 qbtwnre 11904 iooshf 12123 clim2c 14084 lo1o1 14111 climabs0 14164 reef11 14688 absefib 14767 efieq1re 14768 nndivides 14828 oddnn02np1 14910 oddge22np1 14911 evennn02n 14912 evennn2n 14913 halfleoddlt 14924 pc2dvds 15421 pcmpt 15434 subsubc 16336 odmulgid 17794 gexdvds 17822 submcmn2 18067 obslbs 19893 cnntr 20889 cndis 20905 cnindis 20906 cnpdis 20907 lmres 20914 cmpfi 21021 ist0-4 21342 txhmeo 21416 tsmssubm 21756 blin 22036 cncfmet 22519 icopnfcnv 22549 lmmbrf 22868 iscauf 22886 causs 22904 mbfposr 23225 itg2gt0 23333 limcflf 23451 limcres 23456 lhop1 23581 dvdsr1p 23725 fsumvma2 24739 vmasum 24741 chpchtsum 24744 bposlem1 24809 iscgrgd 25208 tgcgr4 25226 lnrot1 25318 eqeelen 25584 nbgraeledg 25959 nb3graprlem2 25981 cusgra3v 25993 cusgrauvtxb 26024 clwlkisclwwlk2 26318 el2spthonot0 26398 rusgranumwlks 26483 dmdmd 28543 funcnvmptOLD 28850 funcnvmpt 28851 1stpreimas 28866 xrdifh 28932 ismntop 29398 eulerpartlemgh 29767 signslema 29965 topdifinfindis 32370 leceifl 32568 lindsenlbs 32574 iblabsnclem 32643 ftc1anclem6 32660 areacirclem5 32674 areacirc 32675 lsatfixedN 33314 cdlemg10c 34945 diaglbN 35362 dih1 35593 dihglbcpreN 35607 mapdcv 35967 ellz1 36348 islssfg 36658 proot1ex 36798 eliooshift 38576 clim2cf 38717 sfprmdvdsmersenne 40058 odd2np1ALTV 40123 nbusgreledg 40575 nb3grprlem2 40609 wspthsnwspthsnon 41122 rusgrnumwwlks 41177 clwlkclwwlk2 41212 |
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