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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 14217 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 = 〈“𝑁𝑂𝑃”〉) |
5 | s4eqd.4 | . . . 4 ⊢ (𝜑 → 𝐷 = 𝑄) | |
6 | 5 | s1eqd 13946 | . . 3 ⊢ (𝜑 → 〈“𝐷”〉 = 〈“𝑄”〉) |
7 | 4, 6 | oveq12d 7153 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) = (〈“𝑁𝑂𝑃”〉 ++ 〈“𝑄”〉)) |
8 | df-s4 14203 | . 2 ⊢ 〈“𝐴𝐵𝐶𝐷”〉 = (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) | |
9 | df-s4 14203 | . 2 ⊢ 〈“𝑁𝑂𝑃𝑄”〉 = (〈“𝑁𝑂𝑃”〉 ++ 〈“𝑄”〉) | |
10 | 7, 8, 9 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 = 〈“𝑁𝑂𝑃𝑄”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 (class class class)co 7135 ++ cconcat 13913 〈“cs1 13940 〈“cs3 14195 〈“cs4 14196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-s1 13941 df-s2 14201 df-s3 14202 df-s4 14203 |
This theorem is referenced by: s5eqd 14219 |
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