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Mirrors > Home > MPE Home > Th. List > oteq2 | Structured version Visualization version 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 4874 | . . 3 ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉) | |
2 | 1 | opeq1d 4879 | . 2 ⊢ (𝐴 = 𝐵 → 〈〈𝐶, 𝐴〉, 𝐷〉 = 〈〈𝐶, 𝐵〉, 𝐷〉) |
3 | df-ot 4637 | . 2 ⊢ 〈𝐶, 𝐴, 𝐷〉 = 〈〈𝐶, 𝐴〉, 𝐷〉 | |
4 | df-ot 4637 | . 2 ⊢ 〈𝐶, 𝐵, 𝐷〉 = 〈〈𝐶, 𝐵〉, 𝐷〉 | |
5 | 2, 3, 4 | 3eqtr4g 2796 | 1 ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 〈cop 4634 〈cotp 4636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-ot 4637 |
This theorem is referenced by: oteq2d 4886 frxp3 8142 xpord3pred 8143 efgi 19635 efgtf 19638 efgtval 19639 |
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