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Mirrors > Home > ILE Home > Th. List > eqbrtri | GIF version |
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
eqbrtr.1 | ⊢ 𝐴 = 𝐵 |
eqbrtr.2 | ⊢ 𝐵𝑅𝐶 |
Ref | Expression |
---|---|
eqbrtri | ⊢ 𝐴𝑅𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrtr.2 | . 2 ⊢ 𝐵𝑅𝐶 | |
2 | eqbrtr.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 2 | breq1i 4007 | . 2 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶) |
4 | 1, 3 | mpbir 146 | 1 ⊢ 𝐴𝑅𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 class class class wbr 4000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 |
This theorem is referenced by: eqbrtrri 4023 3brtr4i 4030 exmidonfinlem 7186 neg1lt0 9013 halflt1 9122 3halfnz 9336 declei 9405 numlti 9406 faclbnd3 10704 geo2lim 11505 0.999... 11510 geoihalfsum 11511 fprodap0 11610 fprodap0f 11625 tan0 11720 cos2bnd 11749 sin4lt0 11755 eirraplem 11765 1nprm 12094 znnen 12379 tan4thpi 13922 zabsle1 14060 ex-fl 14126 trilpolemisumle 14435 |
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