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Mirrors > Home > MPE Home > Th. List > 3bitr3i | Structured version Visualization version GIF version |
Description: A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.) |
Ref | Expression |
---|---|
3bitr3i.1 | ⊢ (𝜑 ↔ 𝜓) |
3bitr3i.2 | ⊢ (𝜑 ↔ 𝜒) |
3bitr3i.3 | ⊢ (𝜓 ↔ 𝜃) |
Ref | Expression |
---|---|
3bitr3i | ⊢ (𝜒 ↔ 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3bitr3i.2 | . . 3 ⊢ (𝜑 ↔ 𝜒) | |
2 | 3bitr3i.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
3 | 1, 2 | bitr3i 277 | . 2 ⊢ (𝜒 ↔ 𝜓) |
4 | 3bitr3i.3 | . 2 ⊢ (𝜓 ↔ 𝜃) | |
5 | 3, 4 | bitri 275 | 1 ⊢ (𝜒 ↔ 𝜃) |
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: an33rean 1484 xorass 1515 cbvaldvaw 2042 sbievw2 2100 sbco4lem 2273 sbco4lemOLD 2274 cbvexv1 2339 cbvex 2399 sbco2d 2512 sbcom 2514 sb7f 2525 eq2tri 2800 clelsb2OLD 2863 clelsb1fw 2908 clelsb1f 2909 cbvraldva 3237 rexcom 3288 cbvrexfw 3303 cbvraldva2 3345 cbvrexdva2OLD 3347 sbralie 3355 ceqsralt 3507 gencbvex 3536 gencbval 3538 ceqsrexbv 3645 ceqsralbv 3646 euind 3721 reuind 3750 sbccomlem 3865 sbccom 3866 csbcom 4418 difcom 4489 eqsn 4833 uniintsn 4992 disjxun 5147 reusv2lem4 5400 exss 5464 opab0 5555 opelinxp 5756 eqbrriv 5792 dm0rn0 5925 elidinxp 6044 qfto 6123 rninxp 6179 coeq0 6255 fununi 6624 dffv2 6987 fndmin 7047 fnprb 7210 fntpb 7211 dfoprab2 7467 frpoins3xp3g 8127 dfer2 8704 eceqoveq 8816 euen1 9026 xpsnen 9055 xpassen 9066 marypha2lem3 9432 rankuni 9858 card1 9963 alephislim 10078 dfacacn 10136 kmlem4 10148 ac6num 10474 zorn2lem4 10494 mappsrpr 11103 sqeqori 14178 trclublem 14942 fprodle 15940 vdwmc2 16912 txflf 23510 metustid 24063 caucfil 24800 ovolgelb 24997 dfcgra2 28081 axcontlem5 28226 frgr3v 29528 nmoubi 30025 hvsubaddi 30319 hlimeui 30493 omlsilem 30655 pjoml3i 30839 hodsi 31028 nmopub 31161 nmfnleub 31178 nmopcoadj0i 31356 pjin3i 31447 or3dir 31702 ralcom4f 31709 rexcom4f 31710 uniinn0 31782 ordtconnlem1 32904 bnj62 33731 bnj610 33758 bnj1143 33801 bnj1533 33863 bnj543 33904 bnj545 33906 bnj594 33923 cusgracyclt3v 34147 xpab 34695 brsuccf 34913 brfullfun 34920 filnetlem4 35266 icorempo 36232 poimirlem13 36501 poimirlem14 36502 poimirlem21 36509 poimirlem22 36510 poimir 36521 sbccom2lem 36992 alrmomorn 37227 dfeldisj5 37591 mpet2 37710 isltrn2N 38991 moxfr 41430 ifporcor 42213 ifpancor 42215 ifpbicor 42226 ifpnorcor 42231 ifpnancor 42232 ifpororb 42256 minregex 42285 relexp0eq 42452 hashnzfzclim 43081 pm11.6 43151 sbc3or 43293 cbvexsv 43308 dfich2 46126 ichbi12i 46128 sprvalpwn0 46151 copisnmnd 46579 |
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