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Mirrors > Home > MPE Home > Th. List > breqi | Structured version Visualization version GIF version |
Description: Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.) |
Ref | Expression |
---|---|
breqi.1 | ⊢ 𝑅 = 𝑆 |
Ref | Expression |
---|---|
breqi | ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breqi.1 | . 2 ⊢ 𝑅 = 𝑆 | |
2 | breq 5149 | . 2 ⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1536 class class class wbr 5147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-cleq 2726 df-clel 2813 df-br 5148 |
This theorem is referenced by: f1ompt 7130 isocnv3 7351 eqfunresadj 7379 brtpos2 8255 brwitnlem 8543 brdifun 8773 omxpenlem 9111 infxpenlem 10050 ltpiord 10924 nqerf 10967 nqerid 10970 ordpinq 10980 ltxrlt 11328 ltxr 13154 trclublem 15030 oduleg 18346 oduposb 18386 join0 18462 meet0 18463 xmeterval 24457 pi1cpbl 25090 slenlt 27811 ltgov 28619 brbtwn 28928 avril1 30491 axhcompl-zf 31026 hlimadd 31221 hhcmpl 31228 hhcms 31231 hlim0 31263 fcoinvbr 32624 brprop 32711 posrasymb 32939 trleile 32945 isarchi 33171 pstmfval 33856 pstmxmet 33857 lmlim 33907 satfbrsuc 35350 brtxp 35861 brpprod 35866 brpprod3b 35868 brtxpsd2 35876 brdomain 35914 brrange 35915 brimg 35918 brapply 35919 brsuccf 35922 brrestrict 35930 brub 35935 brlb 35936 colineardim1 36042 broutsideof 36102 fneval 36334 relowlpssretop 37346 phpreu 37590 poimirlem26 37632 br1cnvres 38250 brid 38287 eqres 38321 alrmomorn 38339 brabidgaw 38346 brabidga 38347 brxrn 38355 br1cossinres 38428 br1cossxrnres 38429 brnonrel 43578 brcofffn 44020 brco2f1o 44021 brco3f1o 44022 clsneikex 44095 clsneinex 44096 clsneiel1 44097 neicvgmex 44106 neicvgel1 44108 climreeq 45568 xlimres 45776 xlimcl 45777 xlimclim 45779 xlimconst 45780 xlimbr 45782 xlimmnfvlem1 45787 xlimmnfvlem2 45788 xlimpnfvlem1 45791 xlimpnfvlem2 45792 xlimuni 45808 gte-lte 48954 gt-lt 48955 gte-lteh 48956 gt-lth 48957 |
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