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Mirrors > Home > ILE Home > Th. List > oteq123d | GIF version |
Description: Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
oteq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
oteq123d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
oteq123d.3 | ⊢ (𝜑 → 𝐸 = 𝐹) |
Ref | Expression |
---|---|
oteq123d | ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐸〉 = 〈𝐵, 𝐷, 𝐹〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oteq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | oteq1d 3582 | . 2 ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐸〉 = 〈𝐵, 𝐶, 𝐸〉) |
3 | oteq123d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | oteq2d 3583 | . 2 ⊢ (𝜑 → 〈𝐵, 𝐶, 𝐸〉 = 〈𝐵, 𝐷, 𝐸〉) |
5 | oteq123d.3 | . . 3 ⊢ (𝜑 → 𝐸 = 𝐹) | |
6 | 5 | oteq3d 3584 | . 2 ⊢ (𝜑 → 〈𝐵, 𝐷, 𝐸〉 = 〈𝐵, 𝐷, 𝐹〉) |
7 | 2, 4, 6 | 3eqtrd 2117 | 1 ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐸〉 = 〈𝐵, 𝐷, 𝐹〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 〈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: (None) |
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