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Mirrors > Home > MPE Home > Th. List > baibd | Structured version Visualization version GIF version |
Description: Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
baibd.1 | ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) |
Ref | Expression |
---|---|
baibd | ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baibd.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) | |
2 | ibar 524 | . . 3 ⊢ (𝜒 → (𝜃 ↔ (𝜒 ∧ 𝜃))) | |
3 | 2 | bicomd 212 | . 2 ⊢ (𝜒 → ((𝜒 ∧ 𝜃) ↔ 𝜃)) |
4 | 1, 3 | sylan9bb 732 | 1 ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 |
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 df-an 385 |
This theorem is referenced by: pw2f1olem 7949 eluz 11577 elicc4 12111 s111 13248 limsupgle 14056 lo1resb 14143 o1resb 14145 isercolllem2 14244 divalgmodcl 14969 ismri2 16115 acsfiel2 16139 issect2 16237 eqglact 17468 eqgid 17469 cntzel 17579 dprdsubg 18246 subgdmdprd 18256 dprd2da 18264 dmdprdpr 18271 issubrg3 18631 ishil2 19882 obslbs 19893 iscld2 20642 isperf3 20767 cncnp2 20895 cnnei 20896 trfbas2 21457 flimrest 21597 flfnei 21605 fclsrest 21638 tsmssubm 21756 isnghm2 22338 isnghm3 22339 isnmhm2 22366 iscfil2 22872 caucfil 22889 ellimc2 23447 cnlimc 23458 lhop1 23581 dvfsumlem1 23593 fsumharmonic 24538 fsumvma 24738 fsumvma2 24739 vmasum 24741 chpchtsum 24744 chpub 24745 rpvmasum2 25001 dchrisum0lem1 25005 dirith 25018 cusgrauvtxb 26024 adjeu 28132 suppss3 28890 nndiffz1 28936 fsumcvg4 29324 qqhval2lem 29353 indpreima 29414 eulerpartlemf 29759 elorvc 29848 neibastop3 31527 relowlpssretop 32388 sstotbnd2 32743 isbnd3b 32754 lshpkr 33422 isat2 33592 islln4 33811 islpln4 33835 islvol4 33878 islhp2 34301 pw2f1o2val2 36625 rfcnpre1 38201 rfcnpre2 38213 uvtx2vtx1edg 40625 uvtx2vtx1edgb 40626 iscplgrnb 40638 frgr3v 41445 |
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