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Mirrors > Home > ILE Home > Th. List > breqtrri | GIF version |
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
breqtrr.1 | ⊢ 𝐴𝑅𝐵 |
breqtrr.2 | ⊢ 𝐶 = 𝐵 |
Ref | Expression |
---|---|
breqtrri | ⊢ 𝐴𝑅𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breqtrr.1 | . 2 ⊢ 𝐴𝑅𝐵 | |
2 | breqtrr.2 | . . 3 ⊢ 𝐶 = 𝐵 | |
3 | 2 | eqcomi 2174 | . 2 ⊢ 𝐵 = 𝐶 |
4 | 1, 3 | breqtri 4014 | 1 ⊢ 𝐴𝑅𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 class class class wbr 3989 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 |
This theorem is referenced by: 3brtr4i 4019 ensn1 6774 pw1dom2 7204 0lt1sr 7727 0le2 8968 2pos 8969 3pos 8972 4pos 8975 5pos 8978 6pos 8979 7pos 8980 8pos 8981 9pos 8982 1lt2 9047 2lt3 9048 3lt4 9050 4lt5 9053 5lt6 9057 6lt7 9062 7lt8 9068 8lt9 9075 nn0le2xi 9185 numltc 9368 declti 9380 sqge0i 10562 faclbnd2 10676 ege2le3 11634 cos2bnd 11723 3dvdsdec 11824 n2dvdsm1 11872 n2dvds3 11874 pockthi 12310 dveflem 13481 tangtx 13553 lgsdir2lem2 13724 ex-fl 13760 |
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