Theorem s3eqd 11303

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 11302	. . 3 |- ( ph -> <" A B "> = <" N O "> )
4		s3eqd.3	. . . 4 |- ( ph -> C = P )
5	4	s1eqd 11153	. . 3 |- ( ph -> <" C "> = <" P "> )
6	3, 5	oveq12d 6019	. 2 |- ( ph -> ( <" A B "> ++ <" C "> ) = ( <" N O "> ++ <" P "> ) )
7		df-s3 11289	. 2 |- <" A B C "> = ( <" A B "> ++ <" C "> )
8		df-s3 11289	. 2 |- <" N O P "> = ( <" N O "> ++ <" P "> )
9	6, 7, 8	3eqtr4g 2287	1 |- ( ph -> <" A B C "> = <" N O P "> )

Syntax hints:   -> wi 4   = wceq 1395  (class class class)co 6001   ++ cconcat 11125   <"cs1 11148   <"cs2 11281   <"cs3 11282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004  df-s1 11149  df-s2 11288  df-s3 11289

This theorem is referenced by:  s4eqd  11304  s3eq2  11309