Theorem s3eqd 11467

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 11466	. . 3 |- ( ph -> <" A B "> = <" N O "> )
4		s3eqd.3	. . . 4 |- ( ph -> C = P )
5	4	s1eqd 11312	. . 3 |- ( ph -> <" C "> = <" P "> )
6	3, 5	oveq12d 6070	. 2 |- ( ph -> ( <" A B "> ++ <" C "> ) = ( <" N O "> ++ <" P "> ) )
7		df-s3 11453	. 2 |- <" A B C "> = ( <" A B "> ++ <" C "> )
8		df-s3 11453	. 2 |- <" N O P "> = ( <" N O "> ++ <" P "> )
9	6, 7, 8	3eqtr4g 2292	1 |- ( ph -> <" A B C "> = <" N O P "> )

Syntax hints:   -> wi 4   = wceq 1398  (class class class)co 6052   ++ cconcat 11282   <"cs1 11307   <"cs2 11445   <"cs3 11446

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3217  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-iota 5314  df-fv 5362  df-ov 6055  df-s1 11308  df-s2 11452  df-s3 11453

This theorem is referenced by:  s4eqd  11468  s3eq2  11473