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Mirrors > Home > MPE Home > Th. List > s4eqd | Structured version Visualization version GIF version |
Description: Equality theorem for a length 4 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) |
s4eqd.4 | ⊢ (𝜑 → 𝐷 = 𝑄) |
Ref | Expression |
---|---|
s4eqd | ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 = 〈“𝑁𝑂𝑃𝑄”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s2eqd.1 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝑁) | |
2 | s2eqd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = 𝑂) | |
3 | s3eqd.3 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝑃) | |
4 | 1, 2, 3 | s3eqd 13460 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 = 〈“𝑁𝑂𝑃”〉) |
5 | s4eqd.4 | . . . 4 ⊢ (𝜑 → 𝐷 = 𝑄) | |
6 | 5 | s1eqd 13234 | . . 3 ⊢ (𝜑 → 〈“𝐷”〉 = 〈“𝑄”〉) |
7 | 4, 6 | oveq12d 6567 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) = (〈“𝑁𝑂𝑃”〉 ++ 〈“𝑄”〉)) |
8 | df-s4 13446 | . 2 ⊢ 〈“𝐴𝐵𝐶𝐷”〉 = (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) | |
9 | df-s4 13446 | . 2 ⊢ 〈“𝑁𝑂𝑃𝑄”〉 = (〈“𝑁𝑂𝑃”〉 ++ 〈“𝑄”〉) | |
10 | 7, 8, 9 | 3eqtr4g 2669 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 = 〈“𝑁𝑂𝑃𝑄”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 (class class class)co 6549 ++ cconcat 13148 〈“cs1 13149 〈“cs3 13438 〈“cs4 13439 |
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 df-s4 13446 |
This theorem is referenced by: s5eqd 13462 |
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