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Mirrors > Home > MPE Home > Th. List > oteq1 | Structured version Visualization version 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 4879 | . . 3 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) | |
2 | 1 | opeq1d 4885 | . 2 ⊢ (𝐴 = 𝐵 → 〈〈𝐴, 𝐶〉, 𝐷〉 = 〈〈𝐵, 𝐶〉, 𝐷〉) |
3 | df-ot 4642 | . 2 ⊢ 〈𝐴, 𝐶, 𝐷〉 = 〈〈𝐴, 𝐶〉, 𝐷〉 | |
4 | df-ot 4642 | . 2 ⊢ 〈𝐵, 𝐶, 𝐷〉 = 〈〈𝐵, 𝐶〉, 𝐷〉 | |
5 | 2, 3, 4 | 3eqtr4g 2791 | 1 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶, 𝐷〉 = 〈𝐵, 𝐶, 𝐷〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 〈cop 4639 〈cotp 4641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-ot 4642 |
This theorem is referenced by: oteq1d 4891 otiunsndisj 5526 frxp3 8165 xpord3pred 8166 efgi 19717 efgtf 19720 efgtval 19721 mapdh9a 41488 mapdh9aOLDN 41489 hdmapval2 41531 otiunsndisjX 46892 |
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