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Theorem s3eqd 14577
Description: Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
s2eqd.1 (𝜑𝐴 = 𝑁)
s2eqd.2 (𝜑𝐵 = 𝑂)
s3eqd.3 (𝜑𝐶 = 𝑃)
Assertion
Ref Expression
s3eqd (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)

Proof of Theorem s3eqd
StepHypRef Expression
1 s2eqd.1 . . . 4 (𝜑𝐴 = 𝑁)
2 s2eqd.2 . . . 4 (𝜑𝐵 = 𝑂)
31, 2s2eqd 14576 . . 3 (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)
4 s3eqd.3 . . . 4 (𝜑𝐶 = 𝑃)
54s1eqd 14306 . . 3 (𝜑 → ⟨“𝐶”⟩ = ⟨“𝑃”⟩)
63, 5oveq12d 7293 . 2 (𝜑 → (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩) = (⟨“𝑁𝑂”⟩ ++ ⟨“𝑃”⟩))
7 df-s3 14562 . 2 ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
8 df-s3 14562 . 2 ⟨“𝑁𝑂𝑃”⟩ = (⟨“𝑁𝑂”⟩ ++ ⟨“𝑃”⟩)
96, 7, 83eqtr4g 2803 1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  (class class class)co 7275   ++ cconcat 14273  ⟨“cs1 14300  ⟨“cs2 14554  ⟨“cs3 14555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-s1 14301  df-s2 14561  df-s3 14562
This theorem is referenced by:  s4eqd  14578  s3eq2  14583  s3sndisj  14678  s3iunsndisj  14679  ragcgr  27068  perpneq  27075  isperp2  27076  isperp2d  27077  footexALT  27079  footexlem2  27081  foot  27083  perprag  27087  perpdragALT  27088  colperpexlem1  27091  lmiisolem  27157  hypcgrlem1  27160  hypcgrlem2  27161  trgcopyeu  27167  iscgra  27170  iscgra1  27171  iscgrad  27172  sacgr  27192  isleag  27208  isleagd  27209  iseqlg  27228
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