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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 13610 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 = 〈“𝑁𝑂𝑃𝑄”〉) |
6 | s5eqd.5 | . . . 4 ⊢ (𝜑 → 𝐸 = 𝑅) | |
7 | 6 | s1eqd 13381 | . . 3 ⊢ (𝜑 → 〈“𝐸”〉 = 〈“𝑅”〉) |
8 | 5, 7 | oveq12d 6668 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸”〉) = (〈“𝑁𝑂𝑃𝑄”〉 ++ 〈“𝑅”〉)) |
9 | df-s5 13596 | . 2 ⊢ 〈“𝐴𝐵𝐶𝐷𝐸”〉 = (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸”〉) | |
10 | df-s5 13596 | . 2 ⊢ 〈“𝑁𝑂𝑃𝑄𝑅”〉 = (〈“𝑁𝑂𝑃𝑄”〉 ++ 〈“𝑅”〉) | |
11 | 8, 9, 10 | 3eqtr4g 2681 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷𝐸”〉 = 〈“𝑁𝑂𝑃𝑄𝑅”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 (class class class)co 6650 ++ cconcat 13293 〈“cs1 13294 〈“cs4 13588 〈“cs5 13589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-s1 13302 df-s2 13593 df-s3 13594 df-s4 13595 df-s5 13596 |
This theorem is referenced by: s6eqd 13612 |
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