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Mirrors > Home > ILE Home > Th. List > oteq2 | GIF version |
Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
Ref | Expression |
---|---|
oteq2 | ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq2 3550 | . . 3 ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉) | |
2 | 1 | opeq1d 3555 | . 2 ⊢ (𝐴 = 𝐵 → 〈〈𝐶, 𝐴〉, 𝐷〉 = 〈〈𝐶, 𝐵〉, 𝐷〉) |
3 | df-ot 3385 | . 2 ⊢ 〈𝐶, 𝐴, 𝐷〉 = 〈〈𝐶, 𝐴〉, 𝐷〉 | |
4 | df-ot 3385 | . 2 ⊢ 〈𝐶, 𝐵, 𝐷〉 = 〈〈𝐶, 𝐵〉, 𝐷〉 | |
5 | 2, 3, 4 | 3eqtr4g 2097 | 1 ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 〈cop 3378 〈cotp 3379 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-ot 3385 |
This theorem is referenced by: oteq2d 3562 |
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