Theorem s3eqd 13609

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

Hypotheses

Ref	Expression
s2eqd.1	|- ( ph -> A = N )
s2eqd.2	|- ( ph -> B = O )
s3eqd.3	|- ( ph -> C = P )

Assertion

Ref	Expression
s3eqd	|- ( ph -> <" A B C "> = <" N O P "> )

Proof of Theorem s3eqd

Step	Hyp	Ref	Expression
1		s2eqd.1	. . . 4 |- ( ph -> A = N )
2		s2eqd.2	. . . 4 |- ( ph -> B = O )
3	1, 2	s2eqd 13608	. . 3 |- ( ph -> <" A B "> = <" N O "> )
4		s3eqd.3	. . . 4 |- ( ph -> C = P )
5	4	s1eqd 13381	. . 3 |- ( ph -> <" C "> = <" P "> )
6	3, 5	oveq12d 6668	. 2 |- ( ph -> ( <" A B "> ++ <" C "> ) = ( <" N O "> ++ <" P "> ) )
7		df-s3 13594	. 2 |- <" A B C "> = ( <" A B "> ++ <" C "> )
8		df-s3 13594	. 2 |- <" N O P "> = ( <" N O "> ++ <" P "> )
9	6, 7, 8	3eqtr4g 2681	1 |- ( ph -> <" A B C "> = <" N O P "> )

Syntax hints:   -> wi 4   = wceq 1483  (class class class)co 6650   ++ cconcat 13293   <"cs1 13294   <"cs2 13586   <"cs3 13587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-s1 13302  df-s2 13593  df-s3 13594

This theorem is referenced by:  s4eqd  13610  s3eq2  13615  s3sndisj  13706  s3iunsndisj  13707  tgcgrxfr  25413  ragcgr  25602  perpneq  25609  isperp2  25610  isperp2d  25611  footex  25613  foot  25614  perprag  25618  perpdragALT  25619  colperpexlem1  25622  lmiisolem  25688  hypcgrlem1  25691  hypcgrlem2  25692  trgcopyeu  25698  iscgra  25701  iscgra1  25702  iscgrad  25703  sacgr  25722  isleag  25733  iseqlg  25747  elwwlks2ons3  26848