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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 14547 | . . 3 ⊢ (𝜑 → 〈“𝐴”〉 = 〈“𝑁”〉) |
3 | s2eqd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = 𝑂) | |
4 | 3 | s1eqd 14547 | . . 3 ⊢ (𝜑 → 〈“𝐵”〉 = 〈“𝑂”〉) |
5 | 2, 4 | oveq12d 7422 | . 2 ⊢ (𝜑 → (〈“𝐴”〉 ++ 〈“𝐵”〉) = (〈“𝑁”〉 ++ 〈“𝑂”〉)) |
6 | df-s2 14795 | . 2 ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) | |
7 | df-s2 14795 | . 2 ⊢ 〈“𝑁𝑂”〉 = (〈“𝑁”〉 ++ 〈“𝑂”〉) | |
8 | 5, 6, 7 | 3eqtr4g 2798 | 1 ⊢ (𝜑 → 〈“𝐴𝐵”〉 = 〈“𝑁𝑂”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 (class class class)co 7404 ++ cconcat 14516 〈“cs1 14541 〈“cs2 14788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-ov 7407 df-s1 14542 df-s2 14795 |
This theorem is referenced by: s3eqd 14811 swrds2m 14888 wrdl2exs2 14893 swrd2lsw 14899 efgi 19580 efgi0 19581 efgi1 19582 efgtf 19583 efgtval 19584 efgval2 19585 frgpuplem 19633 2clwwlk2clwwlklem 29579 wrdt2ind 32095 |
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