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Mirrors > Home > ILE Home > Th. List > breqtrrid | GIF version |
Description: B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.) |
Ref | Expression |
---|---|
breqtrrid.1 | ⊢ 𝐴𝑅𝐵 |
breqtrrid.2 | ⊢ (𝜑 → 𝐶 = 𝐵) |
Ref | Expression |
---|---|
breqtrrid | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breqtrrid.1 | . 2 ⊢ 𝐴𝑅𝐵 | |
2 | breqtrrid.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐵) | |
3 | 2 | eqcomd 2183 | . 2 ⊢ (𝜑 → 𝐵 = 𝐶) |
4 | 1, 3 | breqtrid 4037 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 class class class wbr 4000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 |
This theorem is referenced by: xsubge0 9865 xposdif 9866 bernneq 10623 pcge0 12292 rpabscxpbnd 14019 lgsdir2lem2 14090 trilpolemclim 14433 trilpolemlt1 14438 nconstwlpolemgt0 14460 |
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