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Mirrors > Home > MPE Home > Th. List > syl21anbrc | Structured version Visualization version GIF version |
Description: Syllogism inference. (Contributed by Peter Mazsa, 18-Sep-2022.) |
Ref | Expression |
---|---|
syl21anbrc.1 | ⊢ (𝜑 → 𝜓) |
syl21anbrc.2 | ⊢ (𝜑 → 𝜒) |
syl21anbrc.3 | ⊢ (𝜑 → 𝜃) |
syl21anbrc.4 | ⊢ (𝜏 ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃)) |
Ref | Expression |
---|---|
syl21anbrc | ⊢ (𝜑 → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl21anbrc.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | syl21anbrc.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | syl21anbrc.3 | . . 3 ⊢ (𝜑 → 𝜃) | |
4 | 1, 2, 3 | jca31 514 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∧ 𝜃)) |
5 | syl21anbrc.4 | . 2 ⊢ (𝜏 ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃)) | |
6 | 4, 5 | sylibr 234 | 1 ⊢ (𝜑 → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 |
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 207 df-an 396 |
This theorem is referenced by: fprlem1 8324 erinxp 8830 frrlem15 9795 fpwwe2lem11 10679 nqerf 10968 nqerid 10971 genpcl 11046 nqpr 11052 ltexprlem5 11078 psss 18638 psssdm2 18639 ismhmd 18812 idmhm 18821 resmhm2b 18848 prdspjmhm 18855 pwsdiagmhm 18857 pwsco1mhm 18858 pwsco2mhm 18859 frmdup1 18890 mhmfmhm 19096 isghmd 19256 ghmmhm 19257 idghm 19262 symgsubmefmndALT 19436 lactghmga 19438 frgpmhm 19798 frgpuplem 19805 mulgmhm 19860 isrhm2d 20504 idrhm 20507 pwsco1rhm 20519 pwsco2rhm 20520 subrgid 20590 issubrg2 20609 subsubrg 20615 pwsdiagrhm 20624 islmhmd 21056 reslmhm 21069 rngqiprngho 21331 issubassa 21905 subrgpsr 22016 mat1mhm 22506 mat1rhm 22507 scmatmhm 22556 scmatrhm 22557 mat2pmatmhm 22755 mat2pmatrhm 22756 m2cpmrhm 22768 pm2mpmhm 22842 pm2mprhm 22843 ptpjcn 23635 idnmhm 24791 pi1cpbl 25091 pi1grplem 25096 pi1xfr 25102 pi1coghm 25108 vitalilem1 25657 vitalilem3 25659 ssltd 27851 sssslt1 27855 sssslt2 27856 fldgenfldext 33693 weiunso 36449 prjspertr 42592 prjspvs 42597 0prjspnrel 42614 nla0002 43414 nla0003 43415 clnbgrvtxel 47754 clnbgredg 47764 |
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