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Mirrors > Home > ILE Home > Th. List > oteq1 | GIF version |
Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
Ref | Expression |
---|---|
oteq1 | ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶, 𝐷〉 = 〈𝐵, 𝐶, 𝐷〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 3570 | . . 3 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) | |
2 | 1 | opeq1d 3576 | . 2 ⊢ (𝐴 = 𝐵 → 〈〈𝐴, 𝐶〉, 𝐷〉 = 〈〈𝐵, 𝐶〉, 𝐷〉) |
3 | df-ot 3408 | . 2 ⊢ 〈𝐴, 𝐶, 𝐷〉 = 〈〈𝐴, 𝐶〉, 𝐷〉 | |
4 | df-ot 3408 | . 2 ⊢ 〈𝐵, 𝐶, 𝐷〉 = 〈〈𝐵, 𝐶〉, 𝐷〉 | |
5 | 2, 3, 4 | 3eqtr4g 2138 | 1 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶, 𝐷〉 = 〈𝐵, 𝐶, 𝐷〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 〈cop 3401 〈cotp 3402 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-ot 3408 |
This theorem is referenced by: oteq1d 3582 |
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