Theorem s3eqd 14824

Description: Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
s2eqd.1	⊢ (𝜑 → 𝐴 = 𝑁)
s2eqd.2	⊢ (𝜑 → 𝐵 = 𝑂)
s3eqd.3	⊢ (𝜑 → 𝐶 = 𝑃)

Assertion

Ref	Expression
s3eqd	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)

Proof of Theorem s3eqd

Step	Hyp	Ref	Expression
1		s2eqd.1	. . . 4 ⊢ (𝜑 → 𝐴 = 𝑁)
2		s2eqd.2	. . . 4 ⊢ (𝜑 → 𝐵 = 𝑂)
3	1, 2	s2eqd 14823	. . 3 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)
4		s3eqd.3	. . . 4 ⊢ (𝜑 → 𝐶 = 𝑃)
5	4	s1eqd 14562	. . 3 ⊢ (𝜑 → ⟨“𝐶”⟩ = ⟨“𝑃”⟩)
6	3, 5	oveq12d 7381	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩) = (⟨“𝑁𝑂”⟩ ++ ⟨“𝑃”⟩))
7		df-s3 14809	. 2 ⊢ ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
8		df-s3 14809	. 2 ⊢ ⟨“𝑁𝑂𝑃”⟩ = (⟨“𝑁𝑂”⟩ ++ ⟨“𝑃”⟩)
9	6, 7, 8	3eqtr4g 2800	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)

Syntax hints:   → wi 4   = wceq 1547  (class class class)co 7363   ++ cconcat 14530  ⟨“cs1 14556  ⟨“cs2 14801  ⟨“cs3 14802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-s1 14557  df-s2 14808  df-s3 14809

This theorem is referenced by:  s4eqd  14825  s3eq2  14830  s3sndisj  14927  s3iunsndisj  14928  ragcgr  28800  perpneq  28807  isperp2  28808  isperp2d  28809  footexALT  28811  footexlem2  28813  foot  28815  perprag  28819  perpdragALT  28820  colperpexlem1  28823  lmiisolem  28889  hypcgrlem1  28892  hypcgrlem2  28893  trgcopyeu  28899  iscgra  28902  iscgra1  28903  iscgrad  28904  sacgr  28924  isleag  28940  isleagd  28941  iseqlg  28960