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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 528 | . . 3 ⊢ (𝜒 → (𝜃 ↔ (𝜒 ∧ 𝜃))) | |
3 | 2 | bicomd 223 | . 2 ⊢ (𝜒 → ((𝜒 ∧ 𝜃) ↔ 𝜃)) |
4 | 1, 3 | sylan9bb 509 | 1 ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 |
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 207 df-an 396 |
This theorem is referenced by: rbaibd 540 pw2f1olem 9114 eluz 12889 elicc4 13450 s111 14649 limsupgle 15509 lo1resb 15596 o1resb 15598 isercolllem2 15698 divalgmodcl 16440 ismri2 17676 acsfiel2 17699 eqglact 19209 eqgid 19210 cntzel 19353 dprdsubg 20058 subgdmdprd 20068 dprd2da 20076 dmdprdpr 20083 issubrg3 20616 ishil2 21756 obslbs 21767 iscld2 23051 isperf3 23176 cncnp2 23304 cnnei 23305 trfbas2 23866 flimrest 24006 flfnei 24014 fclsrest 24047 tsmssubm 24166 isnghm2 24760 isnghm3 24761 isnmhm2 24788 iscfil2 25313 caucfil 25330 ellimc2 25926 cnlimc 25937 lhop1 26067 dvfsumlem1 26080 fsumharmonic 27069 fsumvma 27271 fsumvma2 27272 vmasum 27274 chpchtsum 27277 chpub 27278 rpvmasum2 27570 dchrisum0lem1 27574 dirith 27587 uvtx2vtx1edg 29429 uvtx2vtx1edgb 29430 iscplgrnb 29447 frgr3v 30303 adjeu 31917 suppiniseg 32700 suppss3 32741 nndiffz1 32794 islinds5 33374 fsumcvg4 33910 qqhval2lem 33943 indpreima 34005 eulerpartlemf 34351 elorvc 34440 hashreprin 34613 neibastop3 36344 relowlpssretop 37346 sstotbnd2 37760 isbnd3b 37771 lshpkr 39098 isat2 39268 islln4 39489 islpln4 39513 islvol4 39556 islhp2 39979 pw2f1o2val2 43028 modelaxreplem3 44944 rfcnpre1 44956 rfcnpre2 44968 joindm3 48765 meetdm3 48767 catprsc 48801 |
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