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Mirrors > Home > ILE Home > Th. List > breqtri | GIF version |
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
breqtr.1 | ⊢ 𝐴𝑅𝐵 |
breqtr.2 | ⊢ 𝐵 = 𝐶 |
Ref | Expression |
---|---|
breqtri | ⊢ 𝐴𝑅𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breqtr.1 | . 2 ⊢ 𝐴𝑅𝐵 | |
2 | breqtr.2 | . . 3 ⊢ 𝐵 = 𝐶 | |
3 | 2 | breq2i 3995 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶) |
4 | 1, 3 | mpbi 144 | 1 ⊢ 𝐴𝑅𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 class class class wbr 3987 |
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 3587 df-pr 3588 df-op 3590 df-br 3988 |
This theorem is referenced by: breqtrri 4014 3brtr3i 4016 le9lt10 9362 9lt10 9466 sqrt2gt1lt2 11006 trireciplem 11456 cos1bnd 11715 cos2bnd 11716 cos01gt0 11718 sin4lt0 11722 z4even 11868 coseq00topi 13515 sincos4thpi 13520 lgsdir2lem2 13689 lgsdir2lem3 13690 ex-fl 13725 |
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