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| Mirrors > Home > MPE Home > Th. List > oteq3 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
| Ref | Expression |
|---|---|
| oteq3 | ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐷, 𝐴〉 = 〈𝐶, 𝐷, 𝐵〉) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opeq2 4874 | . 2 ⊢ (𝐴 = 𝐵 → 〈〈𝐶, 𝐷〉, 𝐴〉 = 〈〈𝐶, 𝐷〉, 𝐵〉) | |
| 2 | df-ot 4635 | . 2 ⊢ 〈𝐶, 𝐷, 𝐴〉 = 〈〈𝐶, 𝐷〉, 𝐴〉 | |
| 3 | df-ot 4635 | . 2 ⊢ 〈𝐶, 𝐷, 𝐵〉 = 〈〈𝐶, 𝐷〉, 𝐵〉 | |
| 4 | 1, 2, 3 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐷, 𝐴〉 = 〈𝐶, 𝐷, 𝐵〉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 〈cop 4632 〈cotp 4634 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 |
| This theorem is referenced by: oteq3d 4887 otsndisj 5524 otiunsndisj 5525 xpord3pred 8177 efgi0 19738 efgi1 19739 mapdhcl 41729 mapdh6dN 41741 mapdh8 41790 mapdh9a 41791 mapdh9aOLDN 41792 hdmap1l6d 41815 hdmapval 41830 hdmapval2 41834 hdmapval3N 41840 otiunsndisjX 47291 |
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