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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 3931 | . 2 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶) |
4 | 1, 3 | mpbir 145 | 1 ⊢ 𝐴𝑅𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 class class class wbr 3924 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 |
This theorem is referenced by: eqbrtrri 3946 3brtr4i 3953 exmidonfinlem 7042 neg1lt0 8821 halflt1 8930 3halfnz 9141 declei 9210 numlti 9211 faclbnd3 10482 geo2lim 11278 0.999... 11283 geoihalfsum 11284 tan0 11427 cos2bnd 11456 sin4lt0 11462 eirraplem 11472 1nprm 11784 znnen 11900 tan4thpi 12911 ex-fl 12926 trilpolemisumle 13220 |
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