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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 4593 | . . 3 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) | |
2 | 1 | opeq1d 4599 | . 2 ⊢ (𝐴 = 𝐵 → 〈〈𝐴, 𝐶〉, 𝐷〉 = 〈〈𝐵, 𝐶〉, 𝐷〉) |
3 | df-ot 4377 | . 2 ⊢ 〈𝐴, 𝐶, 𝐷〉 = 〈〈𝐴, 𝐶〉, 𝐷〉 | |
4 | df-ot 4377 | . 2 ⊢ 〈𝐵, 𝐶, 𝐷〉 = 〈〈𝐵, 𝐶〉, 𝐷〉 | |
5 | 2, 3, 4 | 3eqtr4g 2858 | 1 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶, 𝐷〉 = 〈𝐵, 𝐶, 𝐷〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 〈cop 4374 〈cotp 4376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-ot 4377 |
This theorem is referenced by: oteq1d 4605 otiunsndisj 5176 efgi 18445 efgtf 18448 efgtval 18449 mapdh9a 37810 mapdh9aOLDN 37811 hdmapval2 37853 otiunsndisjX 42134 |
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