Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 13954 | . . 3 ⊢ (𝜑 → 〈“𝐴”〉 = 〈“𝑁”〉) |
3 | s2eqd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = 𝑂) | |
4 | 3 | s1eqd 13954 | . . 3 ⊢ (𝜑 → 〈“𝐵”〉 = 〈“𝑂”〉) |
5 | 2, 4 | oveq12d 7173 | . 2 ⊢ (𝜑 → (〈“𝐴”〉 ++ 〈“𝐵”〉) = (〈“𝑁”〉 ++ 〈“𝑂”〉)) |
6 | df-s2 14209 | . 2 ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) | |
7 | df-s2 14209 | . 2 ⊢ 〈“𝑁𝑂”〉 = (〈“𝑁”〉 ++ 〈“𝑂”〉) | |
8 | 5, 6, 7 | 3eqtr4g 2881 | 1 ⊢ (𝜑 → 〈“𝐴𝐵”〉 = 〈“𝑁𝑂”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 (class class class)co 7155 ++ cconcat 13921 〈“cs1 13948 〈“cs2 14202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 df-ov 7158 df-s1 13949 df-s2 14209 |
This theorem is referenced by: s3eqd 14225 swrds2m 14302 wrdl2exs2 14307 swrd2lsw 14313 efgi 18844 efgi0 18845 efgi1 18846 efgtf 18847 efgtval 18848 efgval2 18849 frgpuplem 18897 2clwwlk2clwwlklem 28124 wrdt2ind 30627 |
Copyright terms: Public domain | W3C validator |