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| Mirrors > Home > MPE Home > Th. List > s2eqd | Structured version Visualization version GIF version | ||
| Description: Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.) |
| Ref | Expression |
|---|---|
| s2eqd.1 | ⊢ (𝜑 → 𝐴 = 𝑁) |
| s2eqd.2 | ⊢ (𝜑 → 𝐵 = 𝑂) |
| Ref | Expression |
|---|---|
| s2eqd | ⊢ (𝜑 → 〈“𝐴𝐵”〉 = 〈“𝑁𝑂”〉) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | s2eqd.1 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝑁) | |
| 2 | 1 | s1eqd 14639 | . . 3 ⊢ (𝜑 → 〈“𝐴”〉 = 〈“𝑁”〉) |
| 3 | s2eqd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = 𝑂) | |
| 4 | 3 | s1eqd 14639 | . . 3 ⊢ (𝜑 → 〈“𝐵”〉 = 〈“𝑂”〉) |
| 5 | 2, 4 | oveq12d 7449 | . 2 ⊢ (𝜑 → (〈“𝐴”〉 ++ 〈“𝐵”〉) = (〈“𝑁”〉 ++ 〈“𝑂”〉)) |
| 6 | df-s2 14887 | . 2 ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) | |
| 7 | df-s2 14887 | . 2 ⊢ 〈“𝑁𝑂”〉 = (〈“𝑁”〉 ++ 〈“𝑂”〉) | |
| 8 | 5, 6, 7 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → 〈“𝐴𝐵”〉 = 〈“𝑁𝑂”〉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 (class class class)co 7431 ++ cconcat 14608 〈“cs1 14633 〈“cs2 14880 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-s1 14634 df-s2 14887 |
| This theorem is referenced by: s3eqd 14903 swrds2m 14980 wrdl2exs2 14985 swrd2lsw 14991 efgi 19737 efgi0 19738 efgi1 19739 efgtf 19740 efgtval 19741 efgval2 19742 frgpuplem 19790 2clwwlk2clwwlklem 30365 wrdt2ind 32938 elrgspnsubrunlem1 33251 elrgspnsubrun 33253 |
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