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Mirrors > Home > ILE Home > Th. List > impbii | Unicode version |
Description: Infer an equivalence from an implication and its converse. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
impbii.1 |
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impbii.2 |
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Ref | Expression |
---|---|
impbii |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | impbii.1 |
. 2
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2 | impbii.2 |
. 2
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3 | bi3 119 |
. 2
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4 | 1, 2, 3 | mp2 16 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: bicom 140 biid 171 2th 174 pm5.74 179 bitri 184 bibi2i 227 bi2.04 248 pm5.4 249 imdi 250 impexp 263 ancom 266 pm4.45im 334 dfbi2 388 anass 401 pm5.32 453 jcab 603 notnotnot 634 con2b 669 imnan 690 2false 701 pm5.21nii 704 pm4.8 707 oibabs 714 orcom 728 ioran 752 oridm 757 orbi2i 762 or12 766 pm4.44 779 ordi 816 andi 818 pm4.72 827 stdcndc 845 stdcndcOLD 846 stdcn 847 dcnn 848 pm4.82 950 rnlem 976 3jaob 1302 xoranor 1377 falantru 1403 3impexp 1437 3impexpbicom 1438 alcom 1478 19.26 1481 19.3h 1553 19.3 1554 19.21h 1557 19.43 1628 19.9h 1643 excom 1664 19.41h 1685 19.41 1686 equcom 1706 equsalh 1726 equsex 1728 cbvalv1 1751 cbvexv1 1752 cbvalh 1753 cbvexh 1755 sbbii 1765 sbh 1776 equs45f 1802 sb6f 1803 sbcof2 1810 sbequ8 1847 sbidm 1851 sb5rf 1852 sb6rf 1853 equvin 1863 sbimv 1893 cbvalvw 1919 cbvexvw 1920 sbalyz 1999 eu2 2070 eu3h 2071 eu5 2073 mo3h 2079 euan 2082 axext4 2161 cleqh 2277 r19.26 2603 ralcom3 2645 ceqsex 2776 gencbvex 2784 gencbvex2 2785 gencbval 2786 eqvinc 2861 pm13.183 2876 rr19.3v 2877 rr19.28v 2878 reu6 2927 reu3 2928 sbnfc2 3118 difdif 3261 ddifnel 3267 ddifstab 3268 ssddif 3370 difin 3373 uneqin 3387 indifdir 3392 undif3ss 3397 difrab 3410 un00 3470 undifss 3504 ssdifeq0 3506 ralidm 3524 ralf0 3527 ralm 3528 elpr2 3615 snidb 3623 difsnb 3736 preq12b 3771 preqsn 3776 axpweq 4172 exmidn0m 4202 exmidsssn 4203 exmid0el 4205 exmidel 4206 exmidundif 4207 exmidundifim 4208 sspwb 4217 unipw 4218 opm 4235 opth 4238 ssopab2b 4277 elon2 4377 unexb 4443 eusvnfb 4455 eusv2nf 4457 ralxfrALT 4468 uniexb 4474 iunpw 4481 onsucb 4503 unon 4511 sucprcreg 4549 opthreg 4556 ordsuc 4563 dcextest 4581 peano2b 4615 opelxp 4657 opthprc 4678 relop 4778 issetid 4782 xpid11 4851 elres 4944 iss 4954 issref 5012 xpmlem 5050 sqxpeq0 5053 ssrnres 5072 dfrel2 5080 relrelss 5156 fn0 5336 funssxp 5386 f00 5408 f0bi 5409 dffo2 5443 ffoss 5494 f1o00 5497 fo00 5498 fv3 5539 dff2 5661 dffo4 5665 dffo5 5666 fmpt 5667 ffnfv 5675 fsn 5689 fsn2 5691 isores1 5815 ssoprab2b 5932 eqfnov2 5982 cnvoprab 6235 reldmtpos 6254 mapsn 6690 mapsncnv 6695 mptelixpg 6734 elixpsn 6735 ixpsnf1o 6736 en0 6795 en1 6799 dom0 6838 exmidpw 6908 exmidpweq 6909 pw1fin 6910 undifdcss 6922 fidcenum 6955 djuexb 7043 ctssdc 7112 exmidomni 7140 nninfwlpo 7177 exmidfodomr 7203 exmidontri 7238 exmidontri2or 7242 onntri3or 7244 onntri2or 7245 dftap2 7250 exmidmotap 7260 elni2 7313 ltbtwnnqq 7414 enq0ref 7432 elnp1st2nd 7475 elrealeu 7828 elreal2 7829 le2tri3i 8066 elnn0nn 9218 elnnnn0b 9220 elnnnn0c 9221 elnnz 9263 elnn0z 9266 elnnz1 9276 elz2 9324 eluz2b2 9603 elnn1uz2 9607 elpqb 9649 elioo4g 9934 eluzfz2b 10033 fzm 10038 elfz1end 10055 fzass4 10062 elfz1b 10090 fz01or 10111 nn0fz0 10119 fzolb 10153 fzom 10164 elfzo0 10182 fzo1fzo0n0 10183 elfzo0z 10184 elfzo1 10190 rexanuz 10997 rexuz3 10999 sqrt0rlem 11012 fisum0diag 11449 fprod0diagfz 11636 infssuzex 11950 isprm6 12147 oddpwdclemdc 12173 nnoddn2prmb 12262 4sqlem4 12390 ennnfone 12426 ctinfom 12429 ctinf 12431 fnpr2ob 12759 dfgrp2 12902 dfgrp3m 12969 dfgrp3me 12970 isnsg3 13067 dvdsrzring 13496 tgclb 13568 xmetunirn 13861 2sqlem2 14465 bj-nnsn 14488 bdeq 14578 bdop 14630 bdunexb 14675 bj-2inf 14693 bj-nn0suc 14719 exmidsbth 14775 trirec0 14795 redc0 14808 reap0 14809 neap0mkv 14819 |
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