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Mirrors > Home > MPE Home > Th. List > 3bitr3g | Structured version Visualization version GIF version |
Description: More general version of 3bitr3i 289. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.) |
Ref | Expression |
---|---|
3bitr3g.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
3bitr3g.2 | ⊢ (𝜓 ↔ 𝜃) |
3bitr3g.3 | ⊢ (𝜒 ↔ 𝜏) |
Ref | Expression |
---|---|
3bitr3g | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3bitr3g.2 | . . 3 ⊢ (𝜓 ↔ 𝜃) | |
2 | 3bitr3g.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | syl5bbr 273 | . 2 ⊢ (𝜑 → (𝜃 ↔ 𝜒)) |
4 | 3bitr3g.3 | . 2 ⊢ (𝜒 ↔ 𝜏) | |
5 | 3, 4 | syl6bb 275 | 1 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → 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: notbid 307 cador 1538 equequ2OLD 1942 cbvexdva 2271 dfsbcq2 3405 unineq 3836 iindif2 4525 reusv2 4800 rabxfrd 4815 opeqex 4887 eqbrrdv 5140 eqbrrdiv 5141 opelco2g 5211 opelcnvg 5224 ralrnmpt 6276 fliftcnv 6461 eusvobj2 6542 ottpos 7249 smoiso 7346 ercnv 7650 ordiso2 8303 cantnfrescl 8456 cantnfp1lem3 8460 cantnflem1b 8466 cantnflem1 8469 cnfcom 8480 cnfcom3lem 8483 carden2 8696 cardeq0 9253 axpownd 9302 fpwwe2lem9 9339 fzen 12229 hasheq0 13015 incexc2 14409 divalglem4 14957 divalglem8 14961 divalgb 14965 divalgmodOLD 14968 sadadd 15027 sadass 15031 smuval2 15042 smumul 15053 isprm3 15234 vdwmc 15520 imasleval 16024 acsfn2 16147 invsym2 16246 yoniso 16748 pmtrfmvdn0 17705 dprd2d2 18266 cmpfi 21021 xkoinjcn 21300 tgpconcomp 21726 iscau3 22884 mbfimaopnlem 23228 ellimc3 23449 eldv 23468 eltayl 23918 atandm3 24405 el2wlkonot 26396 el2spthonot 26397 rmoxfrdOLD 28716 rmoxfrd 28717 opeldifid 28794 2ndpreima 28868 f1od2 28887 ordtconlem1 29298 bnj1253 30339 wl-dral1d 32497 wl-sb8et 32513 wl-equsb3 32516 wl-sb8eut 32538 wl-ax11-lem8 32548 poimirlem2 32581 poimirlem16 32595 poimirlem18 32597 poimirlem21 32600 poimirlem22 32601 eqbrrdv2 33166 islpln5 33839 islvol5 33883 ntrneicls11 37408 radcnvrat 37535 trsbc 37771 aacllem 42356 |
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