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Mirrors > Home > MPE Home > Th. List > con2bid | Structured version Visualization version GIF version |
Description: A contraposition deduction. (Contributed by NM, 15-Apr-1995.) |
Ref | Expression |
---|---|
con2bid.1 | ⊢ (𝜑 → (𝜓 ↔ ¬ 𝜒)) |
Ref | Expression |
---|---|
con2bid | ⊢ (𝜑 → (𝜒 ↔ ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con2bid.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ ¬ 𝜒)) | |
2 | con2bi 342 | . 2 ⊢ ((𝜒 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜒)) | |
3 | 1, 2 | sylibr 223 | 1 ⊢ (𝜑 → (𝜒 ↔ ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → 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: con1bid 344 sotric 4985 sotrieq 4986 sotr2 4988 isso2i 4991 sotri2 5444 sotri3 5445 somin1 5448 somincom 5449 ordtri2 5675 ordtr3 5686 ordintdif 5691 ord0eln0 5696 iotanul 5783 soisoi 6478 weniso 6504 ordunisuc2 6936 limsssuc 6942 nlimon 6943 tfrlem15 7375 oawordeulem 7521 nnawordex 7604 onomeneq 8035 fimaxg 8092 suplub2 8250 fiming 8287 wemapsolem 8338 cantnflem1 8469 rankval3b 8572 cardsdomel 8683 harsdom 8704 isfin1-2 9090 fin1a2lem7 9111 suplem2pr 9754 xrltnle 9984 ltnle 9996 leloe 10003 xrlttri 11848 xrleloe 11853 xrrebnd 11873 supxrbnd2 12024 supxrbnd 12030 om2uzf1oi 12614 rabssnn0fi 12647 cnpart 13828 bits0e 14989 bitsmod 14996 bitsinv1lem 15001 sadcaddlem 15017 trfil2 21501 xrsxmet 22420 metdsge 22460 ovolunlem1a 23071 ovolunlem1 23072 itg2seq 23315 toslublem 28998 tosglblem 29000 isarchi2 29070 gsumesum 29448 sgnneg 29929 nodenselem7 31086 elfuns 31192 finminlem 31482 bj-bibibi 31744 itg2addnclem 32631 heiborlem10 32789 19.9alt 33270 or3or 37339 ntrclselnel2 37376 clsneifv3 37428 islininds2 42067 |
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