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Theorem s3eqd 13562
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 13561 . . 3 (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)
4 s3eqd.3 . . . 4 (𝜑𝐶 = 𝑃)
54s1eqd 13336 . . 3 (𝜑 → ⟨“𝐶”⟩ = ⟨“𝑃”⟩)
63, 5oveq12d 6633 . 2 (𝜑 → (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩) = (⟨“𝑁𝑂”⟩ ++ ⟨“𝑃”⟩))
7 df-s3 13547 . 2 ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
8 df-s3 13547 . 2 ⟨“𝑁𝑂𝑃”⟩ = (⟨“𝑁𝑂”⟩ ++ ⟨“𝑃”⟩)
96, 7, 83eqtr4g 2680 1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  (class class class)co 6615   ++ cconcat 13248  ⟨“cs1 13249  ⟨“cs2 13539  ⟨“cs3 13540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618  df-s1 13257  df-s2 13546  df-s3 13547
This theorem is referenced by:  s4eqd  13563  s3sndisj  13656  s3iunsndisj  13657  tgcgrxfr  25347  ragcgr  25536  perpneq  25543  isperp2  25544  isperp2d  25545  footex  25547  foot  25548  perprag  25552  perpdragALT  25553  colperpexlem1  25556  lmiisolem  25622  hypcgrlem1  25625  hypcgrlem2  25626  trgcopyeu  25632  iscgra  25635  iscgra1  25636  iscgrad  25637  sacgr  25656  isleag  25667  iseqlg  25681  elwwlks2ons3  26751  frgr2wwlk1  27086  fusgr2wsp2nb  27090
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