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Mirrors > Home > MPE Home > Th. List > s5eqd | Structured version Visualization version GIF version |
Description: Equality theorem for a length 5 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) |
s5eqd.5 | ⊢ (𝜑 → 𝐸 = 𝑅) |
Ref | Expression |
---|---|
s5eqd | ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷𝐸”〉 = 〈“𝑁𝑂𝑃𝑄𝑅”〉) |
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 | s4eqd.4 | . . . 4 ⊢ (𝜑 → 𝐷 = 𝑄) | |
5 | 1, 2, 3, 4 | s4eqd 14901 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 = 〈“𝑁𝑂𝑃𝑄”〉) |
6 | s5eqd.5 | . . . 4 ⊢ (𝜑 → 𝐸 = 𝑅) | |
7 | 6 | s1eqd 14636 | . . 3 ⊢ (𝜑 → 〈“𝐸”〉 = 〈“𝑅”〉) |
8 | 5, 7 | oveq12d 7449 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸”〉) = (〈“𝑁𝑂𝑃𝑄”〉 ++ 〈“𝑅”〉)) |
9 | df-s5 14887 | . 2 ⊢ 〈“𝐴𝐵𝐶𝐷𝐸”〉 = (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸”〉) | |
10 | df-s5 14887 | . 2 ⊢ 〈“𝑁𝑂𝑃𝑄𝑅”〉 = (〈“𝑁𝑂𝑃𝑄”〉 ++ 〈“𝑅”〉) | |
11 | 8, 9, 10 | 3eqtr4g 2800 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷𝐸”〉 = 〈“𝑁𝑂𝑃𝑄𝑅”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 (class class class)co 7431 ++ cconcat 14605 〈“cs1 14630 〈“cs4 14879 〈“cs5 14880 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-s1 14631 df-s2 14884 df-s3 14885 df-s4 14886 df-s5 14887 |
This theorem is referenced by: s6eqd 14903 |
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