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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 14813 | . . 3 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩) |
5 | s4eqd.4 | . . . 4 ⊢ (𝜑 → 𝐷 = 𝑄) | |
6 | 5 | s1eqd 14549 | . . 3 ⊢ (𝜑 → ⟨“𝐷”⟩ = ⟨“𝑄”⟩) |
7 | 4, 6 | oveq12d 7420 | . 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩) = (⟨“𝑁𝑂𝑃”⟩ ++ ⟨“𝑄”⟩)) |
8 | df-s4 14799 | . 2 ⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩) | |
9 | df-s4 14799 | . 2 ⊢ ⟨“𝑁𝑂𝑃𝑄”⟩ = (⟨“𝑁𝑂𝑃”⟩ ++ ⟨“𝑄”⟩) | |
10 | 7, 8, 9 | 3eqtr4g 2789 | 1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ = ⟨“𝑁𝑂𝑃𝑄”⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 (class class class)co 7402 ++ cconcat 14518 ⟨“cs1 14543 ⟨“cs3 14791 ⟨“cs4 14792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-iota 6486 df-fv 6542 df-ov 7405 df-s1 14544 df-s2 14797 df-s3 14798 df-s4 14799 |
This theorem is referenced by: s5eqd 14815 |
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