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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 3997 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶) |
4 | 1, 3 | mpbi 144 | 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: breqtrri 4016 3brtr3i 4018 le9lt10 9369 9lt10 9473 sqrt2gt1lt2 11013 trireciplem 11463 cos1bnd 11722 cos2bnd 11723 cos01gt0 11725 sin4lt0 11729 z4even 11875 coseq00topi 13550 sincos4thpi 13555 lgsdir2lem2 13724 lgsdir2lem3 13725 ex-fl 13760 |
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