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Mirrors > Home > ILE Home > Th. List > eqbrtrrid | GIF version |
Description: B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.) |
Ref | Expression |
---|---|
eqbrtrrid.1 | ⊢ 𝐵 = 𝐴 |
eqbrtrrid.2 | ⊢ (𝜑 → 𝐵𝑅𝐶) |
Ref | Expression |
---|---|
eqbrtrrid | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrtrrid.2 | . 2 ⊢ (𝜑 → 𝐵𝑅𝐶) | |
2 | eqbrtrrid.1 | . 2 ⊢ 𝐵 = 𝐴 | |
3 | eqid 2164 | . 2 ⊢ 𝐶 = 𝐶 | |
4 | 1, 2, 3 | 3brtr3g 4010 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 class class class wbr 3977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2724 df-un 3116 df-sn 3577 df-pr 3578 df-op 3580 df-br 3978 |
This theorem is referenced by: enpr1g 6756 endjudisj 7158 recexprlem1ssl 7566 addgt0 8338 addgegt0 8339 addgtge0 8340 addge0 8341 expge1 10483 expcnv 11435 fprodge1 11570 cos12dec 11698 ncoprmgcdne1b 12010 phicl2 12135 exmidunben 12322 sin0pilem2 13270 cosq23lt0 13321 cos0pilt1 13340 rplogcl 13367 logge0 13368 logdivlti 13369 |
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