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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 205 |
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 206 |
This theorem is referenced by: nbbn 383 pm5.17 1009 dn1 1055 sb8v 2347 sb8f 2348 dfeumo 2530 2ex2rexrot 3294 sbralie 3353 sbralieALT 3354 ceqsralt 3506 reu8 3729 sbcimdv 3851 sbcg 3856 unass 4166 ssin 4230 difab 4300 eq0rdv 4404 csbab 4437 ralidm 4511 iunssf 5047 iunss 5048 poirr 5600 elvvv 5751 cnvuni 5886 dfco2 6244 resin 6855 dffv2 6986 dff1o6 7276 fsplit 8106 naddasslem1 8696 naddasslem2 8697 sbthcl 9098 fiint 9327 rankf 9792 dfac3 10119 dfac5lem3 10123 elznn0 12578 elnn1uz2 12914 lsmspsn 20840 elold 27602 cmbr2i 31117 pjss2i 31201 iuninc 32060 fineqvrep 34394 dffr5 35029 brsset 35166 brtxpsd 35171 ellines 35429 itg2addnclem3 36845 dvasin 36876 cvlsupr3 38518 dihglb2 40517 oneptri 42309 faosnf0.11b 42481 ifpidg 42545 dfsucon 42577 iscard4 42587 dffrege76 42993 dffrege99 43016 ntrneikb 43148 disjinfi 44190 |
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