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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 14228 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 = 〈“𝑁𝑂𝑃”〉) |
5 | s4eqd.4 | . . . 4 ⊢ (𝜑 → 𝐷 = 𝑄) | |
6 | 5 | s1eqd 13957 | . . 3 ⊢ (𝜑 → 〈“𝐷”〉 = 〈“𝑄”〉) |
7 | 4, 6 | oveq12d 7176 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) = (〈“𝑁𝑂𝑃”〉 ++ 〈“𝑄”〉)) |
8 | df-s4 14214 | . 2 ⊢ 〈“𝐴𝐵𝐶𝐷”〉 = (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) | |
9 | df-s4 14214 | . 2 ⊢ 〈“𝑁𝑂𝑃𝑄”〉 = (〈“𝑁𝑂𝑃”〉 ++ 〈“𝑄”〉) | |
10 | 7, 8, 9 | 3eqtr4g 2883 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 = 〈“𝑁𝑂𝑃𝑄”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 (class class class)co 7158 ++ cconcat 13924 〈“cs1 13951 〈“cs3 14206 〈“cs4 14207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-s1 13952 df-s2 14212 df-s3 14213 df-s4 14214 |
This theorem is referenced by: s5eqd 14230 |
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