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Mirrors > Home > ILE Home > Th. List > eqbrtrri | GIF version |
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
eqbrtrr.1 | ⊢ 𝐴 = 𝐵 |
eqbrtrr.2 | ⊢ 𝐴𝑅𝐶 |
Ref | Expression |
---|---|
eqbrtrri | ⊢ 𝐵𝑅𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrtrr.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
2 | 1 | eqcomi 2141 | . 2 ⊢ 𝐵 = 𝐴 |
3 | eqbrtrr.2 | . 2 ⊢ 𝐴𝑅𝐶 | |
4 | 2, 3 | eqbrtri 3944 | 1 ⊢ 𝐵𝑅𝐶 |
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: 3brtr3i 3952 dju1p1e2 7046 expnass 10391 sqrt2gt1lt2 10814 cos1bnd 11455 cos2bnd 11456 2strstr1g 12051 coseq00topi 12905 pigt3 12914 |
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