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Theorem s6eqd 14816
Description: Equality theorem for a length 6 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
s2eqd.1 (𝜑𝐴 = 𝑁)
s2eqd.2 (𝜑𝐵 = 𝑂)
s3eqd.3 (𝜑𝐶 = 𝑃)
s4eqd.4 (𝜑𝐷 = 𝑄)
s5eqd.5 (𝜑𝐸 = 𝑅)
s6eqd.6 (𝜑𝐹 = 𝑆)
Assertion
Ref Expression
s6eqd (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩)

Proof of Theorem s6eqd
StepHypRef Expression
1 s2eqd.1 . . . 4 (𝜑𝐴 = 𝑁)
2 s2eqd.2 . . . 4 (𝜑𝐵 = 𝑂)
3 s3eqd.3 . . . 4 (𝜑𝐶 = 𝑃)
4 s4eqd.4 . . . 4 (𝜑𝐷 = 𝑄)
5 s5eqd.5 . . . 4 (𝜑𝐸 = 𝑅)
61, 2, 3, 4, 5s5eqd 14815 . . 3 (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅”⟩)
7 s6eqd.6 . . . 4 (𝜑𝐹 = 𝑆)
87s1eqd 14549 . . 3 (𝜑 → ⟨“𝐹”⟩ = ⟨“𝑆”⟩)
96, 8oveq12d 7420 . 2 (𝜑 → (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩) = (⟨“𝑁𝑂𝑃𝑄𝑅”⟩ ++ ⟨“𝑆”⟩))
10 df-s6 14801 . 2 ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩)
11 df-s6 14801 . 2 ⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩ = (⟨“𝑁𝑂𝑃𝑄𝑅”⟩ ++ ⟨“𝑆”⟩)
129, 10, 113eqtr4g 2789 1 (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  (class class class)co 7402   ++ cconcat 14518  ⟨“cs1 14543  ⟨“cs5 14793  ⟨“cs6 14794
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  df-s5 14800  df-s6 14801
This theorem is referenced by:  s7eqd  14817
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