![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 276 | . 2 ⊢ (𝜒 ↔ 𝜓) |
4 | 3bitr3i.3 | . 2 ⊢ (𝜓 ↔ 𝜃) | |
5 | 3, 4 | bitri 274 | 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 1483 xorass 1514 cbvaldvaw 2041 sbievw2 2099 sbco4lem 2272 sbco4lemOLD 2273 cbvexv1 2338 cbvex 2398 sbco2d 2511 sbcom 2513 sb7f 2524 eq2tri 2799 clelsb2OLD 2862 clelsb1fw 2907 clelsb1f 2908 cbvraldva 3236 rexcom 3287 cbvrexfw 3302 cbvraldva2 3344 cbvrexdva2OLD 3346 sbralie 3354 ceqsralt 3506 gencbvex 3535 gencbval 3537 ceqsrexbv 3644 ceqsralbv 3645 euind 3720 reuind 3749 sbccomlem 3864 sbccom 3865 csbcom 4417 difcom 4488 eqsn 4832 uniintsn 4991 disjxun 5146 reusv2lem4 5399 exss 5463 opab0 5554 opelinxp 5755 eqbrriv 5791 dm0rn0 5924 elidinxp 6043 qfto 6122 rninxp 6178 coeq0 6254 fununi 6623 dffv2 6986 fndmin 7046 fnprb 7212 fntpb 7213 dfoprab2 7469 frpoins3xp3g 8129 dfer2 8706 eceqoveq 8818 euen1 9028 xpsnen 9057 xpassen 9068 marypha2lem3 9434 rankuni 9860 card1 9965 alephislim 10080 dfacacn 10138 kmlem4 10150 ac6num 10476 zorn2lem4 10496 mappsrpr 11105 sqeqori 14180 trclublem 14944 fprodle 15942 vdwmc2 16914 txflf 23517 metustid 24070 caucfil 24807 ovolgelb 25004 dfcgra2 28119 axcontlem5 28264 frgr3v 29566 nmoubi 30063 hvsubaddi 30357 hlimeui 30531 omlsilem 30693 pjoml3i 30877 hodsi 31066 nmopub 31199 nmfnleub 31216 nmopcoadj0i 31394 pjin3i 31485 or3dir 31740 ralcom4f 31747 rexcom4f 31748 uniinn0 31820 ordtconnlem1 32973 bnj62 33800 bnj610 33827 bnj1143 33870 bnj1533 33932 bnj543 33973 bnj545 33975 bnj594 33992 cusgracyclt3v 34216 xpab 34764 brsuccf 34982 brfullfun 34989 filnetlem4 35352 icorempo 36318 poimirlem13 36587 poimirlem14 36588 poimirlem21 36595 poimirlem22 36596 poimir 36607 sbccom2lem 37078 alrmomorn 37313 dfeldisj5 37677 mpet2 37796 isltrn2N 39077 moxfr 41512 ifporcor 42295 ifpancor 42297 ifpbicor 42308 ifpnorcor 42313 ifpnancor 42314 ifpororb 42338 minregex 42367 relexp0eq 42534 hashnzfzclim 43163 pm11.6 43233 sbc3or 43375 cbvexsv 43390 dfich2 46205 ichbi12i 46207 sprvalpwn0 46230 copisnmnd 46658 |
Copyright terms: Public domain | W3C validator |