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Mirrors > Home > MPE Home > Th. List > 3bitrri | Structured version Visualization version GIF version |
Description: A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.) |
Ref | Expression |
---|---|
3bitri.1 | ⊢ (𝜑 ↔ 𝜓) |
3bitri.2 | ⊢ (𝜓 ↔ 𝜒) |
3bitri.3 | ⊢ (𝜒 ↔ 𝜃) |
Ref | Expression |
---|---|
3bitrri | ⊢ (𝜃 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3bitri.3 | . 2 ⊢ (𝜒 ↔ 𝜃) | |
2 | 3bitri.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
3 | 3bitri.2 | . . 3 ⊢ (𝜓 ↔ 𝜒) | |
4 | 2, 3 | bitr2i 276 | . 2 ⊢ (𝜒 ↔ 𝜑) |
5 | 1, 4 | bitr3i 277 | 1 ⊢ (𝜃 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 |
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 |
This theorem is referenced by: nbbn 383 pm5.17 1013 dn1 1057 sb8v 2353 sb8f 2354 dfeumo 2535 2ex2rexrot 3296 sbralie 3356 sbralieALT 3357 ceqsralt 3514 reu8 3742 sbcimdv 3865 sbcg 3870 unass 4182 ssin 4247 difab 4316 eq0rdv 4413 csbab 4446 ralidm 4518 iunssf 5049 iunss 5050 poirr 5609 elvvv 5764 cnvuni 5900 dfco2 6267 resin 6871 dffv2 7004 dff1o6 7295 fsplit 8141 naddasslem1 8731 naddasslem2 8732 sbthcl 9134 fiint 9364 fiintOLD 9365 rankf 9832 dfac3 10159 dfac5lem3 10163 elznn0 12626 elnn1uz2 12965 lsmspsn 21101 elold 27923 elzs2 28400 cmbr2i 31625 pjss2i 31709 iuninc 32581 fineqvrep 35088 dffr5 35734 brsset 35871 brtxpsd 35876 ellines 36134 itg2addnclem3 37660 dvasin 37691 cvlsupr3 39326 dihglb2 41325 oneptri 43246 faosnf0.11b 43417 ifpidg 43481 dfsucon 43513 iscard4 43523 dffrege76 43929 dffrege99 43952 ntrneikb 44084 disjinfi 45135 |
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