Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4340 | . . 3 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) | |
2 | 1 | opeq1d 4346 | . 2 ⊢ (𝐴 = 𝐵 → 〈〈𝐴, 𝐶〉, 𝐷〉 = 〈〈𝐵, 𝐶〉, 𝐷〉) |
3 | df-ot 4134 | . 2 ⊢ 〈𝐴, 𝐶, 𝐷〉 = 〈〈𝐴, 𝐶〉, 𝐷〉 | |
4 | df-ot 4134 | . 2 ⊢ 〈𝐵, 𝐶, 𝐷〉 = 〈〈𝐵, 𝐶〉, 𝐷〉 | |
5 | 2, 3, 4 | 3eqtr4g 2669 | 1 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶, 𝐷〉 = 〈𝐵, 𝐶, 𝐷〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 〈cop 4131 〈cotp 4133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-ot 4134 |
This theorem is referenced by: oteq1d 4352 otiunsndisj 4905 efgi 17955 efgtf 17958 efgtval 17959 mapdh9a 36097 mapdh9aOLDN 36098 hdmapval2 36142 otiunsndisjX 40317 |
Copyright terms: Public domain | W3C validator |