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Mirrors > Home > MPE Home > Th. List > s3eqd | Structured version Visualization version GIF version |
Description: Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
s2eqd.1 | ⊢ (𝜑 → 𝐴 = 𝑁) |
s2eqd.2 | ⊢ (𝜑 → 𝐵 = 𝑂) |
s3eqd.3 | ⊢ (𝜑 → 𝐶 = 𝑃) |
Ref | Expression |
---|---|
s3eqd | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 = 〈“𝑁𝑂𝑃”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s2eqd.1 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝑁) | |
2 | s2eqd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = 𝑂) | |
3 | 1, 2 | s2eqd 13459 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵”〉 = 〈“𝑁𝑂”〉) |
4 | s3eqd.3 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝑃) | |
5 | 4 | s1eqd 13234 | . . 3 ⊢ (𝜑 → 〈“𝐶”〉 = 〈“𝑃”〉) |
6 | 3, 5 | oveq12d 6567 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) = (〈“𝑁𝑂”〉 ++ 〈“𝑃”〉)) |
7 | df-s3 13445 | . 2 ⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) | |
8 | df-s3 13445 | . 2 ⊢ 〈“𝑁𝑂𝑃”〉 = (〈“𝑁𝑂”〉 ++ 〈“𝑃”〉) | |
9 | 6, 7, 8 | 3eqtr4g 2669 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 = 〈“𝑁𝑂𝑃”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 (class class class)co 6549 ++ cconcat 13148 〈“cs1 13149 〈“cs2 13437 〈“cs3 13438 |
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-rex 2902 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-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-s1 13157 df-s2 13444 df-s3 13445 |
This theorem is referenced by: s4eqd 13461 s3sndisj 13554 s3iunsndisj 13555 tgcgrxfr 25213 ragcgr 25402 perpneq 25409 isperp2 25410 isperp2d 25411 footex 25413 foot 25414 perprag 25418 perpdragALT 25419 colperpexlem1 25422 lmiisolem 25488 hypcgrlem1 25491 hypcgrlem2 25492 trgcopyeu 25498 iscgra 25501 iscgra1 25502 iscgrad 25503 sacgr 25522 isleag 25533 iseqlg 25547 elwwlks2ons3 41159 frgr2wwlk1 41494 fusgr2wsp2nb 41498 |
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